# Bratteli diagrams in Borel dynamics

Sergey Bezuglyi  Department of Mathematics, University of Iowa, Iowa City, USA
<EMAIL_ADDRESS>,  Palle E.T. Jorgensen  Department of Mathematics,
University of Iowa, Iowa City, USA<EMAIL_ADDRESS>,  Olena Karpel
Faculty of Applied Mathematics, AGH University of Krakow, Krakow, Poland & B.
Verkin Institute for Low Temperature Physics and Engineering, Kharkiv, Ukraine
<EMAIL_ADDRESS>and  Shrey Sanadhya  Department of Mathematics, Ben-Gurion
University of the Negev, Beersheba, Israel<EMAIL_ADDRESS>
###### Abstract.

Bratteli-Vershik models have been very successfully applied to the study of
various dynamical systems, in particular, in Cantor dynamics. In this paper,
we study dynamics on the path spaces of generalized Bratteli diagrams that
form models for non-compact Borel dynamical systems. Generalized Bratteli
diagrams have countably infinite many vertices at each level, thus the
corresponding incidence matrices are also countably infinite. We emphasize
differences (and similarities) between generalized and classical Bratteli
diagrams.

Our main results: $(i)$ We utilize Perron-Frobenius theory for countably
infinite matrices to establish criteria for the existence and uniqueness of
tail-invariant path space measures (both probability and $\sigma$-finite).
$(ii)$ We provide criteria for the topological transitivity of the tail
equivalence relation. $(iii)$ We describe classes of stationary generalized
Bratteli diagrams (hence Borel dynamical systems) that: $(a)$ do not support a
probability tail-invariant measure, $(b)$ are not uniquely ergodic with
respect to the tail equivalence relation. $(iv)$ We describe classes of
generalized Bratteli diagrams which can or cannot admit a continuous Vershik
map and construct a Vershik map which is a minimal homeomorphism of a (non
locally compact) Polish space. $(v)$ We provide an application of the theory
of stochastic matrices to analyze diagrams with positive recurrent incidence
matrices.

###### Key words and phrases:

Borel dynamical systems, Bratteli-Vershik model, tail-invariant measures,
infinite ergodic theory.

###### 2020 Mathematics Subject Classification:

37A05, 37B05, 37A40, 54H05, 05C60

###### Contents

  1. 1 Introduction
  2. 2 Generalized Bratteli diagrams: basic definitions and facts
    1. 2.1 Definition of generalized Bratteli diagrams
    2. 2.2 Isomorphism of generalized Bratteli diagrams
  3. 3 Generalized Bratteli diagrams of bounded size
    1. 3.1 The structure of the generalized Bratteli diagrams of bounded size
    2. 3.2 Continuity of a Vershik map for generalized Bratteli diagrams of bounded size
  4. 4 Continuity of Vershik map for generalized Bratteli diagrams
    1. 4.1 Ordered generalized Bratteli diagrams with no infinite minimal and maximal paths.
    2. 4.2 Prolongation of the Vershik map.
  5. 5 Topological transitivity of the tail equivalence relation
  6. 6 Tail-invariant measures for generalized Bratteli diagrams
  7. 7 Uniqueness of tail-invariant measures for stationary Bratteli diagrams
    1. 7.1 Tail-invariant measures and Perron-Frobenius eigenvectors
    2. 7.2 Generalized Bratteli diagrams with finite tail invariant measures
    3. 7.3 Examples
  8. 8 Stochastic matrices in Bratteli diagrams
    1. 8.1 Stochastic matrices and measures
    2. 8.2 Stationary diagrams with positive recurrent incidence matrix
    3. 8.3 Stochastic matrices and measures for non-stationary generalized Bratteli diagrams
  9. 9 Open problems
  10. A Perron-Frobenius theory for infinite matrices
  11. B Examples of Bratteli diagrams supporting invariant measures

##  1\. Introduction

This paper is dedicated to the study of discrete dynamical systems realized on
the path space of generalized Bratteli diagrams. A generalized Bratteli
diagram is a natural extension of the notion of classical (standard) Bratteli
diagrams where each level has a countably infinite set of vertices. The
structure of such diagrams is determined by a sequence of countably infinite
incidence matrices.

In general terms, a Bratteli diagram is a certain combinatorial structure
which encompasses the following: The diagram takes the form of a graph which
is represented as a countable union of levels, and a specification of edges
between levels. The level count in turn is indexed or labeled by non-negative
integers and edges link only levels with index count differing by one. For the
last decades, such diagrams (graphs) have come to serve as a powerful tool for
the analysis of path space, and representing infinite paths via the particular
graphs under consideration. The countable index of the levels may represent
discrete time in associated models of dynamics. Our aim here is to extend
earlier results on dynamics via diagrams, and associated path space
constructions, to the broader context of Borel dynamics, and then to study the
corresponding measures on these generalized path spaces. Our analysis will
entail an extended Perron-Frobenius theory, a tail equivalence relation, and
the construction of corresponding tail-invariant measures, and finally a study
of generalized Vershik maps.

This wide framework of path space analysis is motivated by a variety of
applications. In this paper, we identify new properties of stationary and non-
stationary generalized Bratteli diagrams. Our main results contribute to the
following five important questions: $(i)$ identify the structure and the
properties of Vershik maps and $(ii)$ the tail equivalence relation associated
with generalized Bratteli diagrams; $(iii)$ existence and uniqueness of tail-
invariant measures, finite and $\sigma$-finite. In this context, we present an
analysis of $(iv)$ Bratteli diagrams with positive recurrent incidence
matrices; and $(v)$ path space measures induced by stochastic matrices. These
and other results can be found in Theorems 3.10, 3.12 5.1, 7.2, 7.3, 8.7 and
8.13.

The existing literature on Bratteli diagrams, corresponding dynamical systems,
invariant path-space measures, and other areas used in the paper is very
extensive. We give the references below discussing the most important
ingredients of our work. The reader can find the main ideas in [Bra72],
[HPS92], [GPS95], [Kec95], [Nad90], [Kit98], [Ver82], and other fundamental
works cited below.

Our results may be interesting for mathematicians studying infinite matrices
as well as for experts in Markov chains and random walks on a countable set.
If all incidence matrices of a generalized Bratteli diagram are 0-1 matrices,
then the path space of such a diagram is a well-known object in the theory of
random walks on a countable set. It is worth noting that our focus is on the
study of the tail equivalence relation which gives the principal different
kind of dynamics on the path space. Sections 7 and 8 contain several key
theorems and examples about infinite matrices, their eigenvectors, and
corresponding invariant measures.

Bratteli diagrams, incidence matrices, path spaces. We recall that Bratteli
diagrams are infinite-graded graphs that were named after Ola Bratteli who
introduced them in his pioneering paper [Bra72] on the classification of
approximately finite $C^{*}$-algebras. In short, a Bratteli diagram is a
countable graph $G=(V,E)$ where vertices $V=\bigcup_{n}V_{n}$ and edges
$E=\bigcup_{n}E_{n}$ are divided into disjoint finite sets (levels) $V_{n}$
and $E_{n}$. The edges from the set $E_{n}$ exist only for some vertices from
consecutive levels $V_{n}$ and $V_{n+1}$. This set of edges $E_{n}$ defines a
$|V_{n+1}|\times|V_{n}|$ matrix $F_{n}$ called the incidence matrix. Every
$F_{n}$ has non-negative integer entries. In this paper, we consider the
generalized Bratteli diagrams satisfying the property of finite row sums for
every incidence matrix $F_{n}$. This means that every row has finitely many
non-zero entries. For a Bratteli diagram $B$, the path space $X_{B}$ is formed
by infinite sequences $(e_{i})$ of edges such that $e_{i+1}$ begins at the
vertex where $e_{i}$ ends.

Bratteli diagrams and dynamical systems. For the last decades, Bratteli
diagrams have been intensively studied and used in various areas of dynamics.
The trend to use discrete structures, such as graphs and sequences of
partitions, proved to be a very powerful tool in the theory of dynamical
systems, see e.g. [GM06] and the papers cited below. In the 1970s, Krieger and
Vershik applied sequences of refining partitions to the study of ergodic
automorphisms of a measure space [Ver73], [Ver81], [Ver82], [Kri76], [CK77].
In particular, Vershik proved that any ergodic automorphism of a measure space
can be realized as a transformation acting on the path space indexed by a
sequence of refining partitions, a prototype of a Bratteli diagram.

At the beginning of the 1990s, these ideas found their new applications in
Cantor dynamics. Putnam [Put89] showed that, for every minimal homeomorphism
$\varphi$ of a Cantor set $X$, there exists a sequence of refining partitions
into clopen sets that approximates the orbits of $\varphi$ and the topology on
$X$. Following this article, Herman-Putnam-Skau [HPS92] proved that every
minimal homeomorphism of a Cantor set can be realized as a homeomorphism
$\varphi_{B}$ (called a Vershik map) of a path space $X_{B}$ of a Bratteli
diagram $B$. In other words, Bratteli diagrams represent models of minimal
Cantor dynamical systems. This remarkable result led to a breakthrough in
Cantor dynamics based on the works by Giordano-Putnam-Skau [GPS95]), Glasner-
Weiss [GW95], and others. It was proved that minimal homeomorphisms could be
completely classified with respect to orbit equivalence. The articles by
Forrest [For97] and Durand-Host-Skau [DHS99] answered the question about the
role of stationary Bratteli diagrams: they represent substitution dynamical
systems. Further applications of Bratteli diagrams in Cantor dynamics extended
the class of homeomorphisms that can be realized as Vershik maps. Medynets
[Med06] proved this fact for aperiodic homeomorphisms of a Cantor set (see
also [BDM05]). Recently, the papers by Downarowicz-Karpel [DK19] and Shimomura
[Shi20a], [Shi20b] showed that any homeomorphism of a Cantor set could be
represented as a Vershik map on a Bratteli diagram. There are several
important classes of Bratteli diagrams that deserve special attention.
Stationary non-simple Bratteli diagrams give models for aperiodic substitution
systems [BKM09]). Finite rank Bratteli diagrams (i.e., $|V_{n}|$ is bounded),
which, in particular, represent interval exchange transformations, were
studied in [BKMS13]. Finite and infinite invariant measures were the focus of
the papers [BKK19], [ABKK17]. Eigenvalues of Cantor minimal systems were
considered in a series of papers, see e.g. [DFM19] for references. We do not
discuss all the interesting applications here. The reader can find more
results about invariant measures, dynamics, and applications in [AEG21],
[DP22], [GGS17], [GMPS10], [DM08], [GJ00], [Put18], [Tn18]. See also the
surveys [Dur10], [BK16], [DK18], [BK20] and the literature mentioned there.
The reader who is interested in the classification of stationary Bratteli
diagrams, various links to operator algebras, and $K$-theory can find more
information in [Bra72], [Eff81], [EHS80], [BJKR00], [BJKR01], [BJKR02].

Generalized Bratteli diagrams vs standard Bratteli diagrams. Why do we need
generalized Bratteli diagrams with countable levels? One obvious reason to
study such diagrams is explained by the following result. Bezuglyi-Dooley-
Kwiatkowski [BDK06] proved that every aperiodic Borel automorphism of an
uncountable standard Borel space admits a realization as a Vershik map on the
path space of a generalized Bratteli diagram. A recent result in this
direction was obtained in [BJS22] where the authors proved that there is a
wide class of substitution dynamical systems on infinite alphabets that can be
realized as Vershik maps on stationary generalized Bratteli diagrams. We also
refer our readers to related recent works [MnRW22], [FGMn22]. Among other
possible applications of generalized Bratteli diagrams, we can mention Markov
chains, random walks, iterated function systems [BJ22b], harmonic analysis on
the path space of generalized Bratteli diagrams [BJ21], etc.

It is clear that the variety of classes of generalized Bratteli diagrams is
much wider than that of standard Bratteli diagrams. This fact suggests the
possibility of now establishing key results in the wider context of all
generalized Bratteli diagrams. Another obvious observation is that the case of
countably infinite matrices requires new techniques and methods in comparison
with finite matrices. In this paper, we consider mostly two classes of
diagrams: stationary diagrams (when $F_{n}=F$) and bounded size diagrams (when
all $F_{n}$ are banded matrices with bounded row sums), see Section 2 for
definitions. These diagrams have the predictable behavior of infinite paths
and, therefore, we can better understand their dynamical properties. It is an
interesting problem to find out how much the structure of a generalized
Bratteli diagram determines dynamics and invariant measures on the path space
of a diagram.

The main results of this paper are concentrated on principal problems in
dynamics on the path spaces of Bratteli diagrams. We consider the existence
and uniqueness of measures invariant with respect to the tail equivalence
relation and Vershik maps. We discuss the dynamical properties of the Vershik
map, and how the structure of a Bratteli diagram affects the dynamics on the
path space. The notions of isomorphic and order isomorphic generalized
Bratteli diagrams are considered in this paper.

Extended Perron-Frobenius theory, the tail equivalence relation, and Vershik
map. Our main emphasis in the current work is to point out the differences and
similarities between the dynamics on path space of the classical Bratteli
diagrams and the generalized ones. The tools used in this work are also
significantly different. For example, considering stationary Bratteli
diagrams, we work with Perron-Frobenius eigenpairs. For classical diagrams,
the Perron-Frobenius theory covers all possible cases of stationary Bratteli
diagrams. Moreover, using eigenvectors and eigenvalues of non-negative
matrices, one can explicitly describe all finite and $\sigma$-finite invariant
measures for simple and non-simple stationary Bratteli diagrams [BKMS10]. For
infinite non-negative matrices, the Perron-Frobenius theory does not cover all
cases of generalized Bratteli diagrams: there are stationary diagrams with
incidence matrices that do not have a finite Perron eigenvalue. Another
important circumstance is that the cases of recurrent and transient incidence
matrices lead to essentially different results regarding the uniqueness of
ergodic invariant measures. This means that one needs to use different
techniques for finding invariant measures. We use the book by Kitchens [Kit98]
for references about the definitions and main results of the Perron-Frobenius
theory. For the reader’s convenience, we included in Appendix A the facts that
are used in this paper.

One more essential distinction between standard and generalized Bratteli
diagrams consists of the existence of invariant measures on the path spaces.
For a standard diagram $B$, the path space $X_{B}$ is a compact Cantor set,
and every Vershik map $\varphi_{B}$ is a homeomorphism of $X_{B}$. Therefore,
the classical Bogoliubov-Krylov theorem guarantees the existence of a
probability $\varphi_{B}$-invariant measure on $X_{B}$. By contrast, for a
generalized Bratteli diagram, $X_{B}$ is a Polish zero-dimensional space, and
$(X_{B},\varphi_{B})$ is a Borel dynamical system. There are then generalized
Bratteli diagrams that do not support finite invariant measures. The following
question is natural: for what classes of generalized Bratteli diagrams are
there finite invariant measures? We note that minimal Cantor dynamics deals
with probability invariant measures only. The settings for generalized
Bratteli diagrams lead to the study of both finite and infinite invariant
measures. We construct and describe classes of generalized Bratteli diagrams
that do not support a probability invariant measure.

On every Bratteli diagram $B$, we can consider dynamical systems of two kinds:
the tail equivalence relation $\mathcal{R}$, and a Borel dynamical system
$(X_{B},\varphi_{B})$ defined by a Vershik map $\varphi_{B}$. Then
$\mathcal{R}$ is a countable Borel equivalence relation which is completely
defined by the diagram. To define a Vershik map $\varphi_{B}$, we need to
consider a partial ordering on the set of all edges $E$. The question about
the continuity of a Vershik map was studied in Cantor dynamics [Med06],
[BKY14], [BY17], [JQY17]. The result of Bezuglyi-Dooley-Kwiatkowski [BDK06]
shows that every aperiodic Borel automorphism of an uncountable standard Borel
space admits a realization as a Vershik map on the path space of an ordered
generalized Bratteli diagram such that the Vershik map is a homeomorphism. The
left-to-right ordering on a simple (standard) Bratteli diagram always gives
rise to a continuous Vershik map. But this is not the case for generalized
Bratteli diagrams, even for diagrams with reasonable simple structure, see
Section 3. In other words, there are irreducible generalized Bratteli diagrams
such that the left-to-right order does not generate a continuous Vershik map.
Moreover, in the class of generalized Bratteli diagrams with a unique infinite
minimal path and a unique infinite maximal path one can find diagrams for
which both Vershik map and its inverse are discontinuous, or Vershik map is
continuous but its inverse is not (see Section 4).

Infinite matrices, especially banded matrices, is the subject of great
interest because of their applications in various areas of mathematics and
mathematical physics. In this context we mention [ABS20], [CSZ12], [CSZ13].

The outline of the paper and main results. In Section 2 we provide basic
definitions and discuss the properties of generalized Bratteli diagrams. We
consider such notions as tail equivalence relation, Vershik map, isomorphism
of Bratteli diagrams. In Section 3 we focus mostly on the notion of bounded
size generalized Bratteli diagrams (see Definition 3.1) and show that they
form a natural class of diagrams that can be viewed as intermediate between
classical (standard) and generalized Bratteli diagrams. We show that even
diagrams with this natural structure can provide interesting examples that
contrast the classical case. We find classes of bounded size generalized
Bratteli diagrams such that, for the left-to-right ordering, the Vershik map
is discontinuous (unlike the case of standard Bratteli diagrams), moreover,
every infinite maximal path is a point of discontinuity. We also provide some
classes of bounded size diagrams for which it is possible to prolong the
Vershik map to a homeomorphism. In Section 4, we consider arbitrary
generalized Bratteli diagrams and find conditions under which they have an
order such that the Vershik map can be prolonged to a homeomorphism. In
particular, we are interested in the orders such that the diagram does not
possess infinite minimal and infinite maximal paths. We give sufficient
conditions for a generalized Bratteli diagram to have such an order. We give
an example of an ordered generalized Bratteli diagram with a non locally
compact path space such that the corresponding Vershik map is a minimal
homeomorphism. In contrast to the case of standard Bratteli diagrams, there
are examples of generalized Bratteli diagrams with a unique minimal and a
unique maximal path such that the corresponding Vershik map cannot be
prolonged to a homeomorphism. We show that in the class of ordered generalized
Bratteli diagrams with a unique infinite minimal path and a unique infinite
maximal path one can find examples of diagrams such that (i) both the Vershik
map $\varphi_{B}$ and its inverse $\varphi_{B}^{-1}$ are not continuous; (ii)
the Vershik map $\varphi_{B}$ is continuous but the inverse $\varphi_{B}^{-1}$
is discontinuous; (iii) both the Vershik map $\varphi_{B}$ and its inverse
$\varphi_{B}^{-1}$ are continuous. Section 5 discusses some topological
properties of generalized Bratteli diagrams. It is proved that, for an
irreducible stationary generalized Bratteli diagram, the tail equivalence
relation is topologically transitive. The case of non-stationary diagrams is
also considered. It is proved that irreducible bounded size generalized
Bratteli diagrams do not generate minimal tail equivalence relations. In
Section 6 we discuss the tail invariant measures on the path space of
generalized Bratteli diagrams. We provide explicit examples of non-stationary
generalized Bratteli diagrams that do not support full probability measures
invariant with respect to the Vershik map, and examples of stationary
generalized Bratteli diagrams which do not support finite tail-invariant
measures, i.e., the tail equivalence relations for such diagrams are
compressible. In Section 7, we study finite and $\sigma$-finite invariant
measures on stationary generalized Bratteli diagrams. We give an explicit
description of these measures and prove their uniqueness. Several examples
that illustrate these theorems are also given in Section 7. The detailed
calculations and proofs for these examples are provided in Appendix B. Section
8 deals with applications of stochastic matrices, related to stationary and
non-stationary generalized Bratteli diagrams, to the problem of the existence
of invariant measures. We give examples of null-recurrent incidence matrices
such that the invariant measure for the corresponding generalized Bratteli
diagrams is not unique. We also show that, for positive recurrent matrices,
one can control the growth of the heights of the Kakutani-Rokhlin towers in
terms of the corresponding eigenvectors. In Section 9, we present further
possible directions of research. In Appendix A, we provide a brief description
of the Perron-Frobenius theory for infinite matrices.

##  2\. Generalized Bratteli diagrams: basic definitions and facts

This section contains the main definitions of the objects we consider in the
paper. We discuss generalized Bratteli diagrams and their subclasses, the
corresponding infinite matrices, the tail equivalence relation on the path
space of a diagram, finite and $\sigma$-finite measures invariant with respect
to tail equivalence relation, and orderings on the set of edges and
corresponding Vershik maps.

###  2.1. Definition of generalized Bratteli diagrams

The notion of a generalized Bratteli diagram is a natural extension of the
notion of a Bratteli diagram to the case when all levels in a generalized
Bratteli diagram are countable. It was proved in [BDK06] that any aperiodic
Borel automorphism can be realized as a Vershik map on the path space of a
generalized Bratteli diagram, see Theorem 2.16 below.

We will use the standard notation $\mathbb{N},\mathbb{Z},{\mathbb{R}},$
$\mathbb{N}_{0}=\mathbb{N}\cup\\{0\\}$ for the sets of numbers, and $|\cdot|$
denotes the cardinality of a set.

######  Definition 2.1.

A generalized Bratteli diagram is a graded graph $B=(V,E)$ such that the
vertex set $V$ and the edge set $E$ can be represented as partitions
$V=\bigsqcup_{i=0}^{\infty}V_{i}$ and $E=\bigsqcup_{i=0}^{\infty}E_{i}$
satisfying the following properties:

$(i)$ The number of vertices at each level $V_{i}$, $i\in\mathbb{N}_{0}$ is
countably infinite (in most cases, we will identify each $V_{i}$ with
$\mathbb{Z}$ or $\mathbb{N}$). The set $V_{i}$ is called the $i$th level of
the diagram $B$. For all $i\in\mathbb{N}_{0}$, the set $E_{i}$ of all edges
between $V_{i}$ and $V_{i+1}$ is countable.

$(ii)$ For every edge $e\in E$, we define the range and source maps $r$ and
$s$ such that $r(E_{i})=V_{i+1}$ and $s(E_{i})=V_{i}$ for
$i\in\mathbb{N}_{0}$. In particular, we have $s^{-1}(v)\neq\emptyset$ for all
$v\in V$, and $r^{-1}(v)\neq\emptyset$ for all $v\in V\setminus V_{0}$.

$(iii)$ For every vertex $v\in V\setminus V_{0}$, we have
$|r^{-1}(v)|<\infty$.

######  Remark 2.2.

(1) If the level $V_{0}$ consists of a single vertex and each set $V_{n}$ is
finite, then we get the usual definition of a Bratteli diagram which was first
defined in [Bra72]. In particular, these diagrams were used to model Cantor
dynamical systems and classify them up to orbit equivalence (see [HPS92],
[GPS95]).

(2) Since all levels in a generalized Bratteli diagram are countable sets, we
can use the integers $\mathbb{Z}$ (or natural numbers $\mathbb{N}$) to
enumerate the vertices. When we index the vertices at each level by
$\mathbb{Z}$, the generalized diagram is called two-sided infinite, and when
the vertices are indexed by $\mathbb{N}$, we call it one-sided infinite.

To define the path space of a generalized Bratteli diagram, we consider a
finite or infinite sequence of edges $(e_{i}:e_{i}\in E_{i})$ such that
$s(e_{i})=r(e_{i-1})$ which is called a finite or infinite path, respectively.
Given a generalized Bratteli diagram $B=(V,E)$, we denote the set of infinite
paths starting at $V_{0}$ by $X_{B}$ and call it the path space. For a finite
path $\overline{e}=(e_{0},...,e_{n})$, we denote
$s(\overline{e})=s(e_{0}),r(\overline{e})=r(e_{n})$. The set

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

is called the cylinder set associated with $\overline{e}$.

The topology on the path space $X_{B}$ is generated by cylinder sets. This
topology coincides with the topology defined by the following metric on
$X_{B}$: for $x=(x_{i}),\,y=(y_{i})$, set

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

The path space $X_{B}$ is a zero-dimensional Polish space and therefore a
standard Borel space.

For a vertex $v\in V_{m}$ and a vertex $w\in V_{n}$, we will denote by
$E(v,w)$ the set of all finite paths between $v$ and $w$. Set
$f^{(i)}_{v,w}=|E(v,w)|$ for every $w\in V_{i}$ and $v\in V_{i+1}$. In such a
way, we associate with the generalized Bratteli diagram $B=(V,E)$ a sequence
of non-negative countable infinite matrices $(F_{i})$, $i\in\mathbb{N}_{0}$,
(called the incidence matrices) given by

(2.1) $F_{i}=(f^{(i)}_{v,w}:v\in V_{i+1},w\in V_{i}),\ \
f^{(i)}_{v,w}\in\mathbb{N}_{0}.$

In this paper, a matrix $F=(f_{ij})$ is called infinite (or countably
infinite) if its rows and columns are indexed by the same countably infinite
set. Assuming that all matrices $F^{n},n\in\mathbb{N}$ are defined (i.e., they
have finite entries), we denote the entries of $F^{n}$ by $f_{ij}^{(n)}$.
Observe that this notation is similar to (2.1). It will be clear from the
context whether $f_{ij}^{(n)}$ denotes the $(i,j)$-th entry of the matrix
$F_{n}$ (in a sequence of matrices $(F_{n})_{n\in\mathbb{N}_{0}}$) or it
denotes the $(i,j)$-th entry of the $n$-th power of the matrix $F$.

######  Remark 2.3.

The structure of a generalized Bratteli diagram $B=(V,E)$ is completely
determined by the sequence of incidence matrices
$(F_{n}),\,n\in\mathbb{N}_{0}$. In this case, we write $B=B(F_{n})$. For each
$n\in\mathbb{N}_{0}$, the matrix $F_{n}$ has at most finitely many non-zero
entries in each row, and none of its rows or columns are entirely zero. A
column of $F_{n}$ may have a finite or infinite number of non-zero entries.
Note that, $X_{B}$ is locally compact if and only if every column of $F_{n}$
has finitely many non-zero entries for all $n$.

######  Definition 2.4.

(1) Let $B=B(F_{n})$ be a generalized Bratteli diagram. If $F_{n}=F$ for every
$n\in\mathbb{N}_{0}$, then the diagram $B$ is called stationary. We will write
$B=B(F)$ in this case.

(2) A generalized Bratteli diagram $B=(V,E)$, where all levels $V_{i}$ are
identified with a set $V_{0}$ (e.g. $V_{0}=\mathbb{N}$ or $\mathbb{Z}$), is
called irreducible if for any vertices $i,j\in V_{0}$ and any level $V_{n}$
there exist $m>n$ and a finite path connecting $i\in V_{n}$ and $j\in V_{m}$.
In other words, the $(j,i)$-entry of the matrix $F_{m-1}\cdots F_{n}$ is non-
zero. Otherwise, the diagram is called reducible.

In particular, a generalized stationary Bratteli diagram is irreducible if and
only if the corresponding incidence matrix is irreducible (see Appendix A for
more definitions and results about infinite positive matrices).

######  Definition 2.5.

Two paths $x=(x_{i})$ and $y=(y_{i})$ in $X_{B}$ are called tail equivalent if
there exists an $n\in\mathbb{N}_{0}$ such that $x_{i}=y_{i}$ for all $i\geq
n$. This notion defines a countable Borel equivalence relation $\mathcal{R}$
on the path space $X_{B}$ which is called the tail equivalence relation.

######  Remark 2.6.

The set of generalized Bratteli diagrams contains various ‘‘exotic’’ examples.
In this paper, we will consider the diagrams whose properties are natural from
the point of view of dynamical systems. We will assume that the path space
$X_{B}$ of a generalized Bratteli diagram $B$ has no isolated points, i.e.,
for every infinite path $(x_{0},x_{1},x_{2},...)\in X_{B}$ and every
$n\in\mathbb{N}_{0}$, there exists a level $m>n$ such that
$|s^{-1}(r(x_{m}))|>1$. Hence, the set $X_{B}$ is uncountable. We will
consider only such Bratteli diagrams for which the tail equivalence relation
$\mathcal{R}$ is aperiodic. We will not consider cases where the Bratteli
diagram is a disjoint union of two or more Bratteli diagrams, that is, the
Bratteli diagram should be connected when considered as an undirected graph.

######  Definition 2.7.

Given a generalized Bratteli diagram $B=(V,E)$ and a monotone increasing
sequence $(n_{k}:k\in\mathbb{N}_{0}),n_{0}=0$, we define a new generalized
Bratteli diagram $B^{\prime}=(V^{\prime},E^{\prime})$ as follows: the vertex
sets are determined by $V^{\prime}_{k}=V_{n_{k}}$, and the edge sets
$E^{\prime}_{k}=E_{n_{k}}\circ...\circ E_{n_{k+1}-1}$ are formed by finite
paths between the levels $V^{\prime}_{k}$ and $V^{\prime}_{k+1}$. The diagram
$B^{\prime}=(V^{\prime},E^{\prime})$ is called a telescoping of the original
diagram $B=(V,E)$.

Note that for a generalized stationary Bratteli diagram, after telescoping, we
can always assume that the incidence matrix is aperiodic (see Appendix A).

###  2.2. Isomorphism of generalized Bratteli diagrams

Similarly to the case of standard Bratteli diagrams (see e.g. [Dur10]), we
define the notion of isomorphic generalized Bratteli diagrams:

######  Definition 2.8.

Two generalized Bratteli diagrams $B=(V,E)$ and
$B^{\prime}=(V^{\prime},E^{\prime})$ are called isomorphic if there exist two
sequence of bijections $(g_{n}:V_{n}\rightarrow
V_{n}^{\prime})_{n\in\mathbb{N}_{0}}$ and $(h_{n}:E_{n}\rightarrow
E_{n}^{\prime})_{n\in\mathbb{N}_{0}}$ such that for every
$n\in\mathbb{N}_{0}$, we have $g_{n}(V_{n})=V_{n}^{\prime}$ and
$h_{n}(E_{n})=E_{n}^{\prime}$, and $s^{\prime}\circ h_{n}=g_{n}\circ s$,
$r^{\prime}\circ h_{n}=g_{n}\circ r$.

To illustrate Definition 2.8, we consider the following example.

######  Example 2.9 (Isomorphic generalized Bratteli diagrams, Figure 1).

Let $B(F)$ be a stationary generalized Bratteli diagram such that every level
of $B$ is identified with $\mathbb{Z}$, and the incidence matrix
$F=(f_{ij})_{i,j\in\mathbb{Z}}$ has entries

$f_{ij}=\left\\{\begin{aligned} &2,\mbox{ for }i=j,\\\ &1,\mbox{ for
}|i-j|=1,\\\ &0,\mbox{ otherwise. }\end{aligned}\right.$

Similarly, let $B^{\prime}(F^{\prime})$ be a stationary generalized Bratteli
diagram such that every level of $B^{\prime}$ is identified with
$\mathbb{N}_{0}$, and its incidence matrix
$F^{\prime}=(f^{\prime}_{ij})_{i,j\in\mathbb{N}_{0}}$ has entries
$f^{\prime}_{00}=f^{\prime}_{11}=2$,
$f^{\prime}_{01}=1,f^{\prime}_{02}=1,f^{\prime}_{10}=1,f^{\prime}_{13}=1,f^{\prime}_{20}=1,f^{\prime}_{31}=1$,
and for $i,j\notin\\{0,1\\}$:

$f^{\prime}_{ij}=\left\\{\begin{aligned} &2,\mbox{ for }i=j,\\\ &1,\mbox{ for
}|i-j|=2,\\\ &0,\mbox{ otherwise. }\end{aligned}\right.$

The diagrams $B$ and $B^{\prime}$ (elaborated in Figure 1) are isomorphic.
Indeed, the bijections $(g_{n}:V_{n}\rightarrow
V_{n}^{\prime})_{n\in\mathbb{N}_{0}}$ and $(h_{n}:E_{n}\rightarrow
E_{n}^{\prime})_{n\in\mathbb{N}_{0}}$ that give the isomorphism are defined as
follows: since the diagrams are stationary, we set, for every
$n\in\mathbb{N}_{0}$, $g_{n}=g:\mathbb{Z}\rightarrow\mathbb{N}_{0}$

$g(n)=\left\\{\begin{aligned} &2n,\mbox{ if }n\geq 0,\\\ &-2n-1,\mbox{ if
}n<0.\end{aligned}\right.$

The bijection $h_{n}:E_{n}\rightarrow E_{n}^{\prime}$ is defined as follows:
The two vertical edges in the diagram $B(F)$ with range $i\in V_{n}$ are
mapped to the two vertical edges in the diagram $B^{\prime}(F^{\prime})$ with
range $g(i)\in V_{n}$. For $i>0$, the non-vertical edge with range $i$ coming
from left (respectively from the right) is mapped to the non-vertical edge
with range $g(i)\in V^{\prime}_{n}$ coming from the left (respectively from
the right). For $i<0$, the non-vertical edge with range $i$ coming from left
(respectively from the right) is mapped to the non-vertical edge with range
$g(i)\in V^{\prime}_{n}$ coming from the right (respectively from the left).
For the non-vertical edges with range $0\in V_{n}$, the mapping can be seen
from the Figure 1. It is easy to see that with the above bijections the two
diagrams are isomorphic.

$\ldots$$\ldots$$\ldots$$\ldots$$\vdots$$\vdots$$\vdots$$\vdots$$\vdots$

$\ldots$$\ldots$$\vdots$$\vdots$$\vdots$$\vdots$$\vdots$

Figure 1. Isomorphic generalized Bratteli diagrams $B$ and $B^{\prime}$.

To define a dynamical system on the path space of a generalized Bratteli
diagram, we need to take a linear order $>$ on each (finite) set $r^{-1}(v),\
v\in V\setminus V_{0}$. This order defines a partial order on the sets of
edges $E_{i},\ i=0,1,...$: edges $e,e^{\prime}$ are comparable if and only if
$r(e)=r(e^{\prime})$. For a generalized Bratteli diagram $B=(V,E)$ equipped
with partial order $>$, we define a partial lexicographical order on the set
$E_{k}\circ\cdots\circ E_{l}$ of all finite paths from $V_{k}$ to $V_{l+1}$ as
follows: $(e_{k},...,e_{l})>(f_{k},...,f_{l})$ if and only if $e_{i}>f_{i}$
for some $i$ with $k\leq i\leq l$, and $e_{j}=f_{j}$ for $i<j\leq l$. Then any
two paths from the (finite) set $E(V_{0},v)$ of all paths connecting a vertex
from $V_{0}$ and $v$ are comparable with respect to the lexicographic order.

######  Definition 2.10.

A generalized Bratteli diagram $B=(V,E)$ together with a partial order $>$ on
$E$ is called an ordered generalized Bratteli diagram $B=(V,E,>)$.

We call an infinite path $e=(e_{0},e_{1},...,e_{i},...)$ maximal (respectively
minimal) if every $e_{i}$ has a maximal (respectively minimal) number among
all elements from $r^{-1}(r(e_{i}))$. The same definition is used for finite
maximal/minimal paths. Remark that there are unique minimal and maximal paths
in the set $E(V_{0},v)$ of all finite paths arriving at $v$ for each $v\in
V_{i},\ i>0$.

We note that, in contrast to standard Bratteli diagrams, there are orders on
generalized Bratteli diagrams such that the sets $X_{max}$ and $X_{min}$ of
maximal and minimal paths are empty, see [BDK06] and Example 4.2. It is not
hard to see that $X_{max}$ and $X_{min}$ are closed subsets of $X_{B}$.

######  Definition 2.11.

For an ordered generalized Bratteli diagram $B=(V,E,>)$, we define a Borel
transformation

(2.2) $\varphi_{B}:X_{B}\setminus X_{max}\rightarrow X_{B}\setminus X_{min}$

as follows. Given $x=(x_{0},x_{1},...)\in X_{B}\setminus X_{max}$, let $m$ be
the smallest number such that $x_{m}$ is not maximal. Let $g_{m}$ be the
successor of $x_{m}$ in the finite set $r^{-1}(r(x_{m}))$. Then we set
$\varphi_{B}(x)=(g_{0},g_{1},...,g_{m-1},g_{m},x_{m+1},...)$ where
$(g_{0},g_{1},...,g_{m-1})$ is the minimal path in $E(V_{0},s(g_{m}))$. The
map $\varphi_{B}$ is a Borel bijection. Moreover, $\varphi_{B}$ is a
homeomorphism from $X_{B}\setminus X_{max}$ onto $X_{B}\setminus X_{min}$. If
$\varphi_{B}$ admits a bijective extension to the entire path space $X_{B}$,
then we call the Borel transformation $\varphi_{B}:X_{B}\rightarrow X_{B}$ a
Vershik map, and the Borel dynamical system $(X_{B},\varphi_{B})$ is called a
generalized Bratteli-Vershik system.

######  Remark 2.12.

We collect here several facts about Vershik maps on generalized Bratteli
diagrams.

(1) Let $B=(V,E,>)$ be an ordered generalized Bratteli diagram. Relation (2.2)
defines $\varphi_{B}$ uniquely as a map from $X_{B}\setminus X_{max}$ onto
$X_{B}\setminus X_{min}$. We note that if $\varphi_{B}$ can be prolonged to a
Vershik map on $X_{B}$, then, in general, such an extension is not unique.
Indeed, if $|X_{min}|=|X_{max}|>1$ then we can choose arbitrary a Borel map
from $X_{max}$ onto $X_{min}$ as a Vershik map acting on $X_{max}$.

(2) There exist orders $>$ on $X_{B}$ such that both sets $X_{max}$ and
$X_{min}$ are empty. In this case, $\varphi_{B}$ is uniquely determined
according to (2.2) of Definition 2.11. Also, there orders $>$ on $B$ such that
$|X_{max}|\neq|X_{min}|$; in particular, one of these sets may be empty. In
the latter case, $B(V,E,>)$ does not support a Vershik map.

(3) We note that for simple standard Bratteli diagrams, the left-to-right
ordering always generates a Vershik homeomorphism. Indeed, in this case, all
minimal edges start from the first vertex on each level, and all maximal edges
start from the last vertex. This guarantees the uniqueness of an infinite
minimal and infinite maximal path. For irreducible generalized Bratteli
diagrams the left-to-right ordering does not necessarily produce a Borel
Vershik automorphism (see Example 3.9 below) or a continuous Vershik map (see
Theorem 3.10 below).

######  Definition 2.13.

Two ordered generalized Bratteli diagrams $B=(V,E,>)$,
$B^{\prime}=(V^{\prime},E^{\prime},>^{\prime})$ are called order isomorphic if
they are isomorphic (see Definition 2.8) and, for all $n\in\mathbb{N}$, $v\in
V_{n}$ and $e_{1},e_{2}\in r^{-1}(v)$, we have $e_{2}>e_{1}$ if and only if
$h_{n}(e_{1})>^{\prime}h_{n}(e_{2})$ for $h_{n}(e_{1}),h_{n}(e_{2})\in
r^{-1}(g_{n}(v))$, where $g_{n}:V_{n}\rightarrow V_{n}^{\prime}$ and
$h_{n}:E_{n}\rightarrow E_{n}^{\prime}$ are the bijections defined as in
Definition 2.8.

We show that an order isomorphism of two generalized Bratteli diagrams implies
an isomorphism of the respective generalized Bratteli-Vershik dynamical
systems.

######  Theorem 2.14.

Let $B=(V,E,>)$ and $B^{\prime}=(V^{\prime},E^{\prime},>^{\prime})$ be order
isomorphic generalized Bratteli diagrams. Assume that
$|X_{B}(max)|=|X_{B}(min)|$ and $|X_{B^{\prime}}(max)|=|X_{B^{\prime}}(min)|$.
Then the orders $>$ and $>^{\prime}$ generate Vershik maps
$\varphi_{B}:X_{B}\rightarrow X_{B}$ and
$\varphi_{B^{\prime}}:X_{B^{\prime}}\rightarrow X_{B^{\prime}}$ such that the
generalized Bratteli-Vershik systems $(X_{B},\varphi_{B})$ and
$(X_{B^{\prime}},\varphi_{B^{\prime}})$ are Borel isomorphic.

###### Proof.

We will construct two Vershik maps $\varphi_{B}:X_{B}\rightarrow X_{B}$ and
$\varphi_{B^{\prime}}:X_{B^{\prime}}\rightarrow X_{B^{\prime}}$ and define a
Borel map $f:X_{B}\rightarrow X_{B^{\prime}}$ such that, for $x\in X_{B}$,

(2.3) $f(\varphi_{B}(x))=\varphi_{B^{\prime}}(f(x)).$

We use the concatenation of the maps $(h_{n}:E_{n}\rightarrow
E^{\prime}_{n})_{n\in\mathbb{N}_{0}}$ given in Definition 2.8 to define $f$
for $x=(x_{0},x_{1},...)\in X_{B}$,

$f(x)=(h_{n}(x_{n}))_{n\in\mathbb{N}_{0}}:=(x^{\prime}_{n})_{n\in\mathbb{N}_{0}}:=x^{\prime}\in
X_{B^{\prime}}.$

Since every $h_{n}$ is a bijection, we see that $f:X_{B}\rightarrow
X_{B^{\prime}}$ is a well defined bijective map. It follows from Definition
2.8 that $f(X_{B}(max))=X_{B^{\prime}}(max)$ and
$f(X_{B}(min))=X_{B^{\prime}}(min)$.

By Definition 2.11, we have

(2.4) $\varphi_{B}:X_{B}\setminus X_{B}(max)\rightarrow X_{B}\setminus
X_{B}(min),\ \ \varphi_{B^{\prime}}:X_{B^{\prime}}\setminus
X_{B^{\prime}}(max)\rightarrow X_{B^{\prime}}\setminus X_{B^{\prime}}(min)$

We show that, for $x\in X_{B}\setminus X_{B}(max)$, the map $f:X_{B}\setminus
X_{B}(max)\rightarrow X_{B^{\prime}}\setminus X_{B^{\prime}}(max)$ intertwines
the Vershik maps $\varphi_{B}$ and $\varphi_{B^{\prime}}$. To see this, take
$x=(x_{0},x_{1},...)\in X_{B}\setminus X_{B}(max)$ and find the smallest
integer $k$ such that $x_{k}$ is not maximal. Let $y_{k}$ be the successor of
$x_{k}$ in the finite set $r^{-1}(r(x_{k}))$. Then, by definition of the
Vershik map, $\varphi_{B}(x)=(y_{0},y_{1},...,y_{k-1},y_{k},x_{k+1},...)$
where $(y_{0},y_{1},...,y_{k-1})$ is the minimal path in
$E(V_{0},r(y_{k-1}))$. By definition of $f$, we have

$f(\varphi_{B}(x))=(h_{0}(y_{0}),\cdots,h_{k-1}(y_{k-1}),h_{k}(y_{k}),h_{k+1}(x_{k+1}),\cdots)$

where $(h_{0}(y_{0}),\cdots,h_{k-1}(y_{k-1}))$ is the minimal path with range
$s^{\prime}(h_{k}(y_{k}))$. We note that $h_{k}(y_{k})$ is the successor of
$h_{k}(x_{k})$ in the set $(r^{\prime})^{-1}(r^{\prime}(h(y_{k})))$.

We now compute $\varphi_{B^{\prime}}(f(x))$:

$\displaystyle\varphi_{B^{\prime}}(f(x))=$ $\displaystyle\
\varphi_{B^{\prime}}(h_{0}(x_{0}),\cdots,h_{k-1}(x_{k-1}),h_{k}(x_{k}),h_{k+1}(x_{k+1}),\cdots)$
$\displaystyle=$ $\displaystyle\
(h_{0}(y_{0}),\cdots,h_{k-1}(y_{k-1}),h_{k}(y_{k}),h_{k+1}(x_{k+1}),\cdots)$

because the minimal path with range in $s^{\prime}(h_{k}(y_{k}))$ is unique.
Hence, $\varphi_{B^{\prime}}f=f\varphi_{B}$ on $X_{B}\setminus X_{B}(max)$.

It remains to show that the relation $\varphi_{B^{\prime}}f=f\varphi_{B}$ can
be extended to all $X_{B}$. By the condition of the theorem, $\varphi_{B}$ can
be extended to $X_{B}$, we keep the same notation $\varphi_{B}$ for the
extension. Since $f$ implements a bijection between $X_{B}(max)$ and
$X_{B^{\prime}}(max)$ and between $X_{B}(min)$ and $X_{B^{\prime}}(min)$, we
can extend the definition of $\varphi_{B^{\prime}}$ as follows: for $x\in
X_{B^{\prime}}(max)$, we set

(2.5) $\varphi_{B^{\prime}}(x)=f\circ\varphi_{B}\circ f^{-1}(x).$

Then $\varphi_{B^{\prime}}:X_{B^{\prime}}(max)\to X_{B^{\prime}}(min)$, and,
taking into account the result proved above, we conclude that relation (2.5)
holds for all $x$. ∎

Now we mention an observation that will be of use later on.

######  Lemma 2.15.

Let $B=(V,E,>)$ be an ordered generalized Bratteli diagram. Then the sets
$X_{max}$ and $X_{min}$ of maximal and minimal infinite paths are closed, in
particular, they can be empty. The Vershik map $\varphi_{B}:X_{B}\setminus
X_{max}\rightarrow X_{B}\setminus X_{min}$ is a homeomorphism.

Let $(X,\mathcal{B})$ be a standard Borel space. Recall that any two
uncountable standard Borel spaces are Borel isomorphic. For a standard Borel
space $(X,\mathcal{B})$, a one-to-one Borel map $T$ of $X$ onto itself is
called a Borel automorphism of $(X,\mathcal{B})$. The following result shows
that any aperiodic Borel automorphism of a standard Borel space can be
realized as a Vershik map on a generalized Bratteli diagram.

######  Theorem 2.16 ([BDK06]).

Let $T$ be an aperiodic Borel automorphism acting on a standard Borel space
$(X,\mathcal{B})$. Then there exists an ordered generalized Bratteli diagram
$B=(V,E,>)$ and a Vershik map $\varphi_{B}:X_{B}\to X_{B}$ such that $(X,T)$
is Borel isomorphic to $(X_{B},\varphi_{B})$. Moreover, $\varphi_{B}$ is a
homeomorphism of the path space $X_{B}$.

##  3\. Generalized Bratteli diagrams of bounded size

In this section, we discuss generalized Bratteli diagrams of bounded size (see
Definition 3.1). These diagrams are characterized by the fact that all
incidence matrices are banded. Such diagrams have a locally compact path space
and in our analysis, we consider them as an intermediate step between the
classical Bratteli diagrams (i.e. having finitely many vertices at each level)
and generalized Bratteli diagrams. Generalized Bratteli diagrams of bounded
size present models for substitution dynamical systems on countably infinite
alphabets (see [BJS22]). We present more examples of generalized Bratteli
diagrams of bounded size together with the corresponding substitution
dynamical systems and invariant measures in Subsection 7.3. In Subsection 3.1
we prove statements about the structure of such diagrams which will be of use
later on. In Subsection 3.2, we consider the Vershik map on the path space of
a bounded size diagram defined by the left-to-right order. We show that,
unlike the classical simple Bratteli diagrams, the left-to- right order on an
irreducible generalized Bratteli diagram of bounded size does not necessarily
give rise to a continuous Vershik map. We give conditions under which the
left-to-right order can give rise to a continuous Vershik map and conditions
under which every infinite maximal path is necessarily a point of
discontinuity.

###  3.1. The structure of the generalized Bratteli diagrams of bounded size

This subsection is dedicated to studying the structure of generalized Bratteli
diagrams $B=(V,E)$ of bounded size. Unless stated otherwise, we will identify
the set of vertices at each level with integers, i.e., for each
$i\in\mathbb{N}_{0}$, we have $V_{i}=\mathbb{Z}$. Observe that such a diagram
has a locally compact path space $X_{B}$, and the restriction of $X_{B}$ to
any cylinder set can be represented as the path space of a standard Bratteli
diagram.

######  Definition 3.1.

A generalized Bratteli diagram $B(F_{n})$ is called of bounded size if there
exists a sequence of pairs of natural numbers
$(t_{n},L_{n})_{n\in\mathbb{N}_{0}}$ such that, for all $n\in\mathbb{N}_{0}$
and all $v\in V_{n+1}$,

(3.1)
$s(r^{-1}(v))\in\\{v-t_{n},\ldots,v+t_{n}\\}\quad\mbox{and}\quad\sum_{w\in
V_{n}}f^{(n)}_{vw}=\sum_{w\in V_{n}}|E(w,v)|\leq L_{n}.$

If the sequence $(t_{n},L_{n})_{n\in\mathbb{N}_{0}}$ is constant, i.e.
$t_{n}=t$ and $L_{n}=L$ for all $n\in\mathbb{N}_{0}$, then we say that the
diagram $B(F_{n})$ is of uniformly bounded size.

Observe that, the condition $\sum_{w\in V_{n}}f^{(n)}_{v,w}\leq L_{n}$ implies
that the set $s(r^{-1}(v))$ is finite. Moreover, the cardinality
$|s(r^{-1}(v))|$ is bounded as a function of $v$ for every level $V_{n}$. The
role of the first condition in (3.1) is to control the sources of edges
arriving at $v\in V_{n+1}$.

######  Remark 3.2.

(1) We will use the following convention for bounded size Bratteli diagrams.
For each $n\in\mathbb{N}_{0}$, the pair of natural numbers $(t_{n},L_{n})$ are
chosen to be minimal possible. Also, for every $n\in\mathbb{N}_{0}$, it is
assumed that $E(v-t_{n},v)$ and $E(v+t_{n},v)$ are nonempty for all $v\in
V_{n+1}$. These assumptions are made to simplify our notation and
calculations. Otherwise, we would have to use two sequences, $(t_{n}^{+})$,
$(t_{n}^{-})$, where

$t_{n}^{+}=\max\\{t:E(v+t,v)\neq\emptyset\\}$

and

$t_{n}^{-}=\max\\{t:E(v-t,v)\neq\emptyset\\}.$

The usage of two different sequence $(t^{\pm}_{n})$ instead of $(t_{n})$ would
affect computations but not the corresponding results.

(2) Observe that the property of bounded size implies that the incidence
matrices of the diagram are banded infinite matrices. Let $B=B(F_{n})$ be a
generalized Bratteli diagram of bounded size corresponding to a sequence
$(t_{n},L_{n})_{n\in\mathbb{N}_{0}}$. Then all non-zero entries of $F_{n}$
belong to a band of width $2t_{n}+1$ along the main diagonal. Moreover, the
sum of entries in every row of $F_{n}$ is bounded by $L_{n}$. In other words,
for every $n\in\mathbb{N}_{0}$ and $v\in V_{n+1}$ we have $f^{(n)}_{v,w}=0\
\mbox{if}\ |v-w|>t_{n}$ and $\sum_{w\in V_{n}}f^{(n)}_{v,w}\leq L_{n}$. Also,
by the above assumption, $E(v\pm t_{n},v)\neq\emptyset$ (or $f^{(n)}_{v,v\pm
t_{n}}>0$) for all $v\in V_{n+1},v\pm t_{n}\in V_{n}$ and
$n\in\mathbb{N}_{0}$.

In [BJS22], the authors studied substitution dynamical systems on a countably
infinite alphabet ${\mathcal{A}}$ as Borel dynamical systems. It was proved
that a substitution dynamical system on a countably infinite alphabet which is
left determined (a generalization of the recognizability property to the
countable case, see [Fer06]) and has bounded size (see the definition below)
admits a realization as a Vershik map on a stationary generalized Bratteli
diagram.

######  Definition 3.3.

Identifying ${\mathcal{A}}$ with $\mathbb{Z}$, we say that a substitution
$\sigma:n\to\sigma(n)$, $n\in\mathbb{Z}$, is of bounded size if it is of
bounded length and there exists a positive integer $t$ (independent of $n$)
such that for every $n\in\mathbb{Z}$, if $m\in\sigma(n)$, then
$m\in\\{n-t,...,n,...,n+t\\}$.

Clearly, this definition is analogous to the definition of bounded size
generalized Bratteli diagrams.

######  Theorem 3.4.

[BJS22] Let $\sigma$ be a bounded size left determined substitution on a
countably infinite alphabet and $(X_{\sigma},T)$ be the corresponding
subshift. Then there exists a stationary ordered generalized Bratteli diagram
$B=(V,E,\geq)$ of bounded size and a Vershik map $\varphi\colon
X_{B}\rightarrow X_{B}$ such that $(X_{\sigma},T)$ is Borel isomorphic to
$(X_{B},\varphi)$.

In the rest of the subsection, we discuss the structure of generalized
Bratteli diagrams of bounded size.

######  Lemma 3.5.

Let $B=(V,E)$ be a generalized Bratteli diagram of bounded size. Let
$n\in\mathbb{N}_{0}$, $v\in V_{n+1}$ and $E(V_{0},v)$ be the set of all finite
paths $\overline{e}=(e_{0},\ldots,e_{n})$ such that $r(\overline{e})=v$. Then

$s(E(V_{0},v))\subset\left\\{v-\sum_{i=0}^{n}t_{i},\ldots,v+\sum_{i=0}^{n}t_{i}\right\\}$

and

$|E(V_{0},v)|\leq L_{0}\cdots L_{n}.$

###### Proof.

We prove the lemma by induction. Case $n=0$ follows from the definition.
Suppose the statement of the lemma is true for $n=k$. Then for any $v\in
V_{k+2}$,

$s(r^{-1}(v))\in\\{v-t_{k+1},\ldots,v+t_{k+1}\\}$

and

$\displaystyle s(E(V_{0},v))$ $\displaystyle\subset$
$\displaystyle\bigcup_{w\in\\{v-t_{k+1},\ldots,v+t_{k+1}\\}}s(E(V_{0},w))$
$\displaystyle\subset$
$\displaystyle\left\\{v-t_{k+1}-\sum_{i=0}^{k}t_{i},\ldots,v+t_{k+1}+\sum_{i=0}^{k}t_{i}\right\\}$
$\displaystyle=$
$\displaystyle\left\\{v-\sum_{i=0}^{k+1}t_{i},\ldots,v+\sum_{i=0}^{k+1}t_{i}\right\\}.$

We also have

$|E(V_{0},v)|=\sum_{w\in V_{k+1}}f_{vw}^{(k+1)}|E(V_{0},w)|\leq(L_{0}\cdots
L_{k})\sum_{w\in V_{k+1}}f_{vw}^{(k+1)}\leq L_{0}\cdots L_{k}\cdot L_{k+1}.$

∎

######  Corollary 3.6.

Let $B=(V,E)$ be a generalized Bratteli diagram of bounded size. Let
$n\in\mathbb{N}_{0}$, $v\in V_{n+1}$ and $E(V_{0},v)$ be the set of all finite
paths $\overline{e}=(e_{0},\ldots,e_{n})$ such that $r(\overline{e})=v$. Then
for every $m\leq n$ we have

$s(E(V_{m},v))\subset\left\\{v-\sum_{i=m}^{n}t_{i},\ldots,v+\sum_{i=m}^{n}t_{i}\right\\}.$

###### Proof.

The proof follows from an induction argument similar to the proof of Lemma
3.5. ∎

Assume that $B=(V,E)$ is a diagram of bounded size corresponding to a sequence
$(t_{n},L_{n})_{n\in\mathbb{N}_{0}}$. Fix $v\in V_{n+1}$ and let
$\overline{e}=(e_{0},\ldots,e_{n})$ be a finite path with
$r(\overline{e})=v\in V_{n+1}$. By Lemma 3.5,

$s(e_{0})\in\left\\{v-\sum_{i=0}^{n}t_{i},\ldots,v+\sum_{i=0}^{n}t_{i}\right\\}.$

######  Lemma 3.7.

Let $B=(V,E)$ be a diagram of bounded size corresponding to a sequence
$(t_{n},L_{n})_{n\in\mathbb{N}_{0}}$. Fix $v\in V_{n+1}$, and let
$\overline{e}=(e_{0},\ldots,e_{n})$ be a finite path with
$r(\overline{e})=v\in V_{n+1}$. Then for all infinite paths
$x=(x_{n})_{n\in\mathbb{N}_{0}}\in[\overline{e}]$ and all $m\geq 0$:

(3.2)
$r(x_{n+m})\in\left\\{v-\sum_{i=1}^{m}t_{n+i},\ldots,v+\sum_{i=1}^{m}t_{n+i}\right\\}\subset
V_{n+m+1}.$

###### Proof.

We prove this lemma by induction. Fix $v\in V_{n+1}$, then (3.2) is trivially
true for $m=0$. By induction step, we assume that (3.2) holds for $m=k$ i.e.

$r(x_{n+k})\in\left\\{v-\sum_{i=1}^{k}t_{n+i},\ldots,v+\sum_{i=1}^{k}t_{n+i}\right\\}\subset
V_{n+k+1}.$

By Lemma 3.5 if $u\in V_{n+k+2}$, such that $E(v,u)\neq\emptyset$ then

$v\in\left\\{u-\sum_{i=0}^{k}t_{n+1+i},\ldots,u+\sum_{i=0}^{k}t_{n+1+i}\right\\}.$

This implies,

$u\in\left\\{v-\sum_{i=1}^{k+1}t_{n+i},\ldots,v+\sum_{i=1}^{k+1}t_{n+i}\right\\}.$

In other words

$r(x_{n+k+1})\in\left\\{v-\sum_{i=1}^{k+1}t_{n+i},\ldots,v+\sum_{i=1}^{k+1}t_{n+i}\right\\}\subset
V_{n+k+2}$

as needed. ∎

###  3.2. Continuity of a Vershik map for generalized Bratteli diagrams of
bounded size

In the case of classical Bratteli diagrams, i.e., the diagrams with finitely
many vertices at each level, it is a well-known fact that, for simple Bratteli
diagrams, the left-to-right order always gives rise to a Vershik homeomorphism
[HPS92], [Dur10]. In this subsection, we show that this is not true for
irreducible generalized Bratteli diagrams of bounded size.

Let $B=(V,E)$ be an irreducible generalized Bratteli diagram of bounded size
with the corresponding sequence $(t_{n},L_{n})_{n\in\mathbb{N}_{0}}$. We will
denote by $\omega$ a fixed partial order on $E$. To emphasize that a diagram
$B$ is ordered, we will write $(B,\omega)$. Let $X_{max}=X_{max}(\omega)$ and
$X_{min}=X_{min}(\omega)$ denote the sets of infinite maximal and minimal
paths, respectively, with respect to the corresponding order.

######  Lemma 3.8.

Let $B=(B,\omega)$ be an ordered generalized Bratteli diagram of bounded size
where $\omega$ is the left-to-right partial order on $E$, then

$|X_{max}|=|X_{min}|=\aleph_{0}.$

###### Proof.

Let $(t_{n},L_{n})_{n\in\mathbb{N}_{0}}$ denote the sequence corresponding to
the bounded size diagram $(B,\omega)$. Recall that for $n\in\mathbb{N}$ and
$v\in V_{n}$, $E(V_{0},v)$ denotes the set of all finite paths
$\overline{e}=(e_{0},\ldots,e_{n})$ such that $r(\overline{e})=v$. Since
$\omega$ is left-to-right ordering, the leftmost finite path and the rightmost
finite path is the unique minimal and the unique maximal path, respectively,
in the set $E(V_{0},v)$. We denote them by $\overline{e}_{min}$ and
$\overline{e}_{max}$. Since $(B,\omega)$ is of bounded size there are vertices
$u=v+t_{n+1}$ and $u^{\prime}=v-t_{n+1}$ in $V_{n+1}$ such that $e(v,u)$ is
the minimal edge in the set $r^{-1}(u)$ and $e(v,u^{\prime})$ is the maximal
edge in the set $r^{-1}(u^{\prime})$.

This observation implies that, for every $v\in V_{n}$, the finite minimal path
$\overline{e}_{min}\in E(V_{0},v)$ and the finite maximal path
$\overline{e}_{max}\in E(V_{0},v))$ have unique minimal and unique maximal
extensions to the level $n+1$. This proves that the sets $X_{max}$ and
$X_{min}$ are countable. ∎

Observe that for a two-sided generalized Bratteli diagram of bounded size
$B=(V,E)$ (i.e. each $V_{i}$ is identified with $\mathbb{Z}$) with left-to-
right ordering there exist countably infinite ‘‘slanted’’ minimal paths which
go ‘‘from left-to-right’’ countably infinite ‘‘slanted’’ maximal paths which
go ‘‘from right to left’’. Since by convention, each vertex has the rightmost
and leftmost incoming edge corresponding to the parameters $t_{n}$, all
infinite minimal paths go ‘‘parallelly’’ to each other, the same holds for
maximal paths.

Note that for an ordered generalized Bratteli diagram $(B,\omega)$, the
Vershik map $\varphi_{B}(\omega)$ (corresponding to $\omega$) is a well
defined map from $X_{B}\setminus X_{max}(\omega)$ to $X_{B}\setminus
X_{min}(\omega)$. Denote by $\Phi_{B}(\omega)$ the set of all possible
extensions of $\varphi_{B}(\omega)$ to Vershik maps defined on the entire
$X_{B}$. In general, $\Phi_{B}(\omega)$ can be empty. As mentioned in Lemma
2.15, the map $\varphi_{B}(\omega):X_{B}\setminus X_{max}(\omega)\to
X_{B}\setminus X_{min}(\omega)$ is continuous. In what follows, we discuss the
question of whether $\varphi_{B}(\omega)$ can be extended to a continuous
Vershik map on the entire path space $X_{B}$.

We first give a simple example where there is no Vershik map on a generalized
Bratteli diagram, in other words, the map $\varphi_{B}(\omega):X_{B}\setminus
X_{max}(\omega)\to X_{B}\setminus X_{min}(\omega)$ can not be extended to the
entire path space in Borel fashion.

######  Example 3.9.

Let $B=(B,\omega)$ be a one-sided generalized Bratteli diagram of bounded
sized (i.e. each $V_{i}$ is identified with $\mathbb{N}$) where $\omega$ is
the left-to-right ordering. We considered such a diagram in Example 2.9
(although without the ordering, see the second diagram in Figure 1). Then it
is easy to see that $(B,\omega)$ admits infinitely many minimal paths and no
maximal paths in the diagram. Hence, the Vershik map
$\varphi_{B}:X_{B}\setminus X_{max}\to X_{B}\setminus X_{min}$ cannot be
extended to a Borel bijection to $X_{B}$.

Now we show that there exists a class of generalized Bratteli diagrams of
bounded size such that the left-to-right order does not give rise to a
continuous Vershik map.

######  Theorem 3.10.

Let $B=(B,\omega)$ be a generalized Bratteli diagram of bounded size, with the
corresponding sequence $(t_{n},L_{n})_{n\in\mathbb{N}_{0}}$ and left-to-right
ordering $\omega$. Assume that for all $N\in\mathbb{N}$ there exists
$k\in\mathbb{N}$ such that

(3.3) $t_{N+k}<\sum_{i=N}^{N+k-1}t_{i}.$

Let $\varphi_{B}(\omega)\in\Phi(\omega)$ be any Vershik map on $X_{B}$
corresponding to the order $\omega$. Then $\varphi_{B}(\omega)$ is not
continuous. Moreover, every infinite maximal path is a point of discontinuity.

###### Proof.

Assume that the vertices of $B$ are enumerated by $\mathbb{Z}$. By Lemma 3.8,
there always exists a well defined Vershik map
$\varphi_{B}=\varphi_{B}(\omega)$ on $X_{B}$. To show that $\varphi_{B}$ is
not continuous, we will prove that any neighborhood of a maximal path contains
two infinite paths, $x^{\prime}$ and $x^{\prime\prime}$, such that the
distance between their images under the Vershik map is equal to $1$.

Pick any $x=(x_{n})_{n\in\mathbb{N}_{0}}\in X_{max}$, we will show that in an
arbitrarily small neighborhood of $x$ there are paths
$x^{\prime}=(x^{\prime}_{n})_{n\in\mathbb{N}_{0}},x^{\prime\prime}=(x^{\prime\prime}_{n})_{n\in\mathbb{N}_{0}}\in
X_{B}\setminus X_{max}$ such that
$d(\varphi_{B}(x^{\prime}),\varphi_{B}(x^{\prime\prime})=1$. For
$\varepsilon>0$, take any $n$ such that $\frac{1}{2^{n-1}}<\varepsilon$. Let
$x^{\prime}_{l}=x_{l}$ for $l<n$ and $x_{n}^{\prime}$ be the minimal edge with
the source in $r(x_{n-1})$. Define $x^{\prime}_{k}$ for $k>n$ in an arbitrary
way such that $x^{\prime}=(x_{l}^{\prime})$ form a path in $X_{B}$. Then
$d(x,x^{\prime})<\varepsilon$ and $x^{\prime}$ is not a maximal path. Let
$y^{\prime}=(y^{\prime}_{n})=\varphi_{B}(x^{\prime})$. Then $y^{\prime}_{n}$
is the successor of $x_{n}^{\prime}$ and since the ordering is left-to-right,
we have $s(y_{n}^{\prime})\geq r(x_{n-1})=s(x^{\prime}_{n})$.

By (3.3), there exists $k\in\mathbb{N}$ such that

$t_{n+k}<\sum_{i=n}^{n+k-1}t_{i}.$

Set $x^{\prime\prime}_{l}=x_{l}$ for $0\leq l\leq n+k-1$. Let
$x_{n+k}^{\prime\prime}$ be the minimal edge with the source in $r(x_{n+k-1})$
and let $x^{\prime\prime}_{l}$ for $l>n+k-1$ be defined in an arbitrary way
such that $x^{\prime\prime}=(x_{l}^{\prime\prime})$ form a path in $X_{B}$.
Then $d(x,x^{\prime\prime})<\varepsilon$ and $x^{\prime\prime}$ is not a
maximal path. Let $y^{\prime\prime}_{n+k}$ be the successor of
$x^{\prime\prime}_{n+k}$ and denote
$y^{\prime\prime}=(y^{\prime\prime}_{l})=\varphi_{B}(x^{\prime\prime})$. Since
$B$ is a diagram of bounded size, we have $s(y^{\prime\prime}_{n+k})\leq
r(x_{n+k-1})+2t_{n+k}$. Recall that we assume that the sets $E(v-t_{n},v)$ and
$E(v+t_{n},v)$ are nonempty for all $v\in V_{n+1}$ and all $n$. This means
that

$r(x_{n+k-1})=r(x_{n-1})-\sum_{i=n}^{n+k-1}t_{i}.$

Since $(y_{0}^{\prime\prime},\ldots,y_{n+k-1}^{\prime\prime})$ is the finite
minimal path and the sets $E(v-t_{n},v)$, $E(v+t_{n},v)$ are nonempty for
every $v\in V_{n+1}$, we have

$s(y^{\prime\prime}_{n})=s(y^{\prime\prime}_{n+k})-\sum_{i=n}^{n+k-1}t_{i}.$

Thus,

$\displaystyle s(y^{\prime\prime}_{n})=$ $\displaystyle
s(y^{\prime\prime}_{n+k})-\sum_{i=n}^{n+k-1}t_{i}$ $\displaystyle\leq$
$\displaystyle r(x_{n+k-1})+2t_{n+k}-\sum_{i=n}^{n+k-1}t_{i}$ $\displaystyle=$
$\displaystyle r(x_{n-1})+2t_{n+k}-2\sum_{i=n}^{n+k-1}t_{i}$ $\displaystyle<$
$\displaystyle r(x_{n-1}).$

Hence, $s(y_{n}^{\prime})>s(y_{n}^{\prime\prime})$ and since all minimal paths
in the diagram go ‘‘parallelly’’ to each other, we have
$d(\varphi_{B}(x^{\prime}),\varphi_{B}(x^{\prime\prime}))=1$. ∎

######  Remark 3.11.

We note that the conditions of the theorem above hold for generalized Bratteli
diagrams of uniformly bounded size. Indeed, since $t_{n}=t$ for all
$n\in\mathbb{N}_{0}$, for every $N\in\mathbb{N}$ it is enough to take $k=2$
and the condition (3.3) will be satisfied. Thus, even irreducible generalized
Bratteli diagrams of uniformly bounded size with left-to-right ordering do not
support a continuous Vershik map.

Now we show that there exists a class of generalized Bratteli diagrams of
bounded size such that the left-to-right order does give rise to a continuous
Vershik map. This class is dual to the class of diagrams considered in Theorem
3.10.

######  Theorem 3.12.

Let $B=(B,\omega)$ be a generalized Bratteli diagram of bounded size, with the
corresponding sequence $(t_{n},L_{n})_{n\in\mathbb{N}_{0}}$ and left-to-right
ordering $\omega$. Let $L_{n}=2$ for all $n\in\mathbb{N}$ and let there exist
$N\in\mathbb{N}_{0}$ such that

(3.4) $t_{N+k}=\sum_{i=N}^{N+k-1}t_{i}$

for all $k\in\mathbb{N}$. Then there exists a continuous Vershik map
corresponding to $\omega$.

###### Proof.

Clearly, the Vershik map is well-defined on $X_{B}\setminus X_{max}$. Define
the Vershik map $\varphi_{B}$ on the set of maximal paths as follows: for
every vertex $w\in V_{N}$, let $\varphi_{B}$ map the unique maximal path
$x_{max}^{(w)}$ passing through $w$ to the unique minimal path $x_{min}^{(w)}$
passing through $w$. Then $\varphi_{B}$ is a bijection. Since every vertex
$v\in V\setminus V_{0}$ has exactly two incoming edges, it is easy to verify
the continuity of $\varphi_{B}$. Indeed, let $w\in V_{N}$ and $x$ be any non-
maximal path that coincides with $x_{max}^{(w)}$ up to level $M\geq N$. Then

$d(x_{max}^{(w)},x)=\frac{1}{2^{M}}\leq\frac{1}{2^{N}}.$

Then it follows from the structure of the diagram that the distance

$d(\varphi_{B}(x),x^{(w)}_{min})=\frac{1}{2^{M}}.$

Thus, $x$ will be mapped to the neighborhood of the corresponding $x_{min}$
(see Figure 2 in Example 3.13 below). ∎

$\ldots$$\ldots$$\ldots$$\ldots$$\ldots$$\ldots$$\ldots$$\ldots$$\vdots$$\vdots$$\vdots$$\vdots$$\vdots$
Figure 2. A continuous Vershik map (left-to-right ordering)

Now we give an example which illustrates Theorem 3.12.

######  Example 3.13.

Let $B=(B,\omega)$ be a two-sided generalized Bratteli diagram of bounded
size, with the corresponding sequence $(t_{n},L_{n})_{n\in\mathbb{N}_{0}}$ and
left-to-right ordering $\omega$. Suppose that

$L_{n}=2,\,\,\,t_{n}=2^{n-1}\,\,\,\mathrm{for}\,\,\,n\in\mathbb{N}\,\,\,\mathrm{and}\,\,\,t_{0}=1.$

The diagram $B=(V,E)$ is shown in Figure 2. In other words, every vertex $v\in
V\setminus V_{0}$ has exactly two incoming edges. Endow $B$ with the left-to-
right ordering. To define the Vershik map, we use the following rule: for
every vertex $w\in V_{0}$, the maximal infinite path which starts at $w$ is
mapped to the minimal infinite path which starts at the same vertex $w$. Then
it is easy to check that the corresponding Vershik map is a homeomorphism.
Note that, in this case, for every $n\geq 1$

$t_{n}=\sum_{i=0}^{n-1}t_{i}$

(compare with the result of Theorem 3.12). We remark also that the Bratteli
diagram, constructed in this example, is reducible. Indeed, starting from the
level $V_{2}$, all infinite paths that begin at even vertices will go through
even vertices only.

##  4\. Continuity of Vershik map for generalized Bratteli diagrams

The aim of this section is to find conditions under which a generalized
Bratteli diagram admits an order such that the corresponding Vershik map can
be prolonged to a homeomorphism. In particular, we are interested in orders
$\omega$ such that there are no infinite minimal and no infinite maximal
paths. The Vershik map $\varphi_{B}(\omega)$ corresponding to such an order is
uniquely defined, and it is always a homeomorphism. Such orders were also used
in Theorem 2.16 where generalized Bratteli diagrams corresponding to Borel
dynamical systems were constructed using sequences of vanishing markers
[BDK06].

###  4.1. Ordered generalized Bratteli diagrams with no infinite minimal and
maximal paths.

Let $B$ be a generalized Bratteli diagram such that, for every level
$n\in\mathbb{N}$, the set of its vertices $V_{n}$ is identified with
$\mathbb{N}$. Note that this condition is not a restriction, since the set of
vertices of any generalized Bratteli diagram can be enumerated this way (see
Subsection 2.2 and Example 2.9). We call an edge $e$ of such a diagram slanted
from right to left if $r(e)<s(e)$, slanted from left to right if $r(e)>s(e)$,
and vertical if $r(e)=s(e)$. We also say that a finite or infinite path is
slanted from right to left if it consists of edges slanted from right to left.

######  Theorem 4.1.

Let $B$ be a generalized Bratteli diagram such that the set of vertices on
each level is identified with $\mathbb{N}$. Suppose that there exists a level
$N$ such that starting from that level every vertex has at least two incoming
edges which are slanted from right to left. Then $B$ admits an order such that
the sets of infinite minimal and infinite maximal paths are empty.

###### Proof.

Put an arbitrary order on $B$ up to level $N$. Starting from level $N$, we
define an order such that, for every vertex, all minimal and maximal edges are
slanted from right to left. Indeed, by the assumption, every vertex $v$
beginning level $N$ and below has at least two incoming edges slanted from
right to left, say $e_{v}$ and $e^{\prime}_{v}$. We define an order $\omega$
such that a minimal edge and a maximal edge in $r^{-1}(v)$ are $e_{v}$ and
$e^{\prime}_{v}$, respectively. Since all vertices are enumerated by natural
numbers, we denote by $v_{1}$ the leftmost vertex in every level starting from
$N$. Then the vertex $v_{1}$ of the diagram has no outgoing minimal edge and
no outgoing maximal edge since all its outgoing edges are either vertical or
slanted from left to right.

Let $x=(x_{n})$ be any minimal or maximal path that passes through some vertex
$w\in V_{N}$. Since for every $n\geq N$, all minimal and maximal edges are
slanted from right to left, we obtain $r(x_{n})<w+N-n$ and
$r(x_{n+1})<r(x_{n})$ for every $n\geq N$. Since all levels of vertices are
enumerated by natural numbers, every such path $x$ must be finite.

∎

######  Example 4.2.

The diagram in Figure 3 satisfies the conditions of Theorem 4.1 for $N=0$.
Indeed, define $B=B(F)$ to be a one-sided infinite generalized Bratteli
diagram where $A=F^{T}=(a_{ij})_{i,j\in\mathbb{N}}$ is defined by

(4.1) $A=\begin{pmatrix}1&1&1&1&1&1&\cdots\\\ 1&0&0&0&0&0&\cdots\\\
1&1&0&0&0&0&\cdots\\\ 0&1&1&0&0&0&\cdots\\\ 0&0&1&1&0&0&\cdots\\\
0&0&0&1&1&0&\cdots\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\ddots\end{pmatrix}.$

In other words, for all $j\in\mathbb{N}$,

(4.2) $a_{ij}=\left\\{\begin{aligned} &1\mbox{ if }i=1\mbox{ or }i=j+1\mbox{
or }i=j+2,\\\ &0\mbox{ otherwise }.\end{aligned}\right.$

Note that every vertex in $V\setminus V_{0}$ has exactly three incoming edges.
Define a stationary order on $B$ as follows: let all the edges outgoing from
the first vertex have label $1$, and all other edges can be labeled by $0$ and
$2$ in an arbitrary way. So it defines an ordered generalized Bratteli
diagram. Then, for any vertex $v\in V_{0}$, any minimal or maximal path
starting from this vertex will be slanted from right to left and finite (see
Figure 3 below). Notice that the Vershik map corresponding to this diagram is
a minimal homeomorphism. Indeed, every infinite path in $X_{B}$ passes through
the first vertex infinitely many times and the orbit of such path visits every
cylinder set of $X_{B}$.

$1$$1$$\ldots$$\ldots$$\ldots$$\vdots$$\vdots$$\vdots$$\vdots$$\vdots$$0$$2$$0$$2$$0$$2$$0$$2$$0$$2$$0$$2$$1$$1$$1$$1$
Figure 3. The Vershik map is a minimal homeomorphism, the sets of infinite
minimal and maximal paths are empty

######  Example 4.3.

The diagram in Figure 4 satisfies the conditions of the Theorem 4.1 for $N=0$
after telescoping with respect to even levels. The incidence matrix of the
diagram has the form

$F=\begin{pmatrix}2&1&0&0&0&\cdots\\\ 1&2&1&0&0&\cdots\\\ 0&1&2&1&0&\cdots\\\
0&0&1&2&1&\cdots\\\ 0&0&0&1&2&\cdots\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\ddots\end{pmatrix}.$

The orders of $E_{0}$ and $E_{1}$ are presented in Figure 4. For every even
$i$, the edges $E_{i}$ are enumerated in the same way as $E_{0}$, and for
every odd $i$, in the same way as $E_{1}$. Thus, after telescoping with
respect to even-numbered levels, we obtain a stationary ordered Bratteli
diagram such that all minimal and maximal edges are slanted from right to
left.

210231023102312310102210321032103210$\vdots$$\vdots$$\vdots$$\vdots$$\vdots$$\ldots$$\ldots$$\ldots$
Figure 4. The Vershik map is a homeomorphism: after telescoping, all minimal
and maximal edges are slanted from right to left.

######  Remark 4.4.

Note that if every vertex of a generalized Bratteli diagram has an outgoing
minimal and an outgoing maximal edge then the sets of infinite minimal paths
and infinite maximal paths are non-empty. The diagram in Example 4.3 has the
property that every vertex has at least one extreme (minimal or maximal)
outgoing edge, but this does not guarantee the existence of infinite maximal
or infinite minimal paths. Moreover, the aforementioned property does not hold
after telescoping with respect to even levels. After such a telescoping, the
first vertex does not have any outgoing minimal or maximal edges.

######  Example 4.5.

The diagram shown in Figure 5 does not satisfy conditions of Theorem 4.1 for
any $N$, but still, there is an order for which there are no infinite minimal
and infinite maximal paths. For every vertex $w\in V\setminus V_{0}$, if there
are two incoming edges that are slanted from right to left, we enumerate them
as minimal and maximal. Otherwise, we enumerate two vertical incoming edges as
minimal and maximal. The remaining edges can be enumerated in arbitrary ways,
for instance, from left to right. Then every minimal and maximal edge is
either vertical or slanted from right to left and any minimal or maximal path
eventually goes only through edges slanted from right to left.

This example shows that the assumption made in Theorem 4.1 is not necessary.

$2$$1$$2$$0$$1$$0$$1$$0$$0$$3$$2$$1$$3$$2$$0$$3$$2$$0$$1$$0$$0$$4$$\vdots$$\vdots$$\vdots$$\vdots$$\ldots$$\ldots$$\ldots$$\ldots$$2$$1$$0$$3$$3$$2$$0$$4$$0$$4$$2$$3$$0$$2$
Figure 5. The sets of infinite minimal and maximal paths are empty (the labels
are shown only for bow-shaped edges).

######  Remark 4.6.

Not every generalized Bratteli diagram can be endowed with an order such that
there are no infinite minimal and no infinite maximal paths. It is easy to
provide examples of reducible generalized Bratteli diagrams such that for any
order there exist infinite minimal and infinite maximal paths. We recall the
definition of a vertex subdiagram which was used before for standard Bratteli
diagrams but can be also defined in the same way for generalized ones. Let
$B=(V,E)$ be a generalized Bratteli diagram. Let
$\overline{W}=\\{W_{n}\\}_{n>0}$ be a sequence of proper, non-empty subsets
$W_{n}\subset V_{n}$. Set $W^{\prime}_{n}=V_{n}\setminus W_{n}$. The (vertex)
subdiagram $\overline{B}=(\overline{W},\overline{E})$ is a (standard or
generalized) Bratteli diagram defined by the vertices
$\overline{W}=\bigcup_{i\geq 0}W_{n}$ and the edges $\overline{E}$ that have
their source and range in $\overline{W}$. In other words, the incidence matrix
$\overline{F}_{n}$ of $\overline{B}$ is defined by those edges from $B$ that
have their source and range in vertices from $W_{n}$ and $W_{n+1}$,
respectively (see e.g. [BK16]). Suppose that a path space of a generalized
Bratteli diagram $B$ has a compact subset that is invariant under the tail
equivalence relation $\mathcal{R}$ and is represented by a standard Bratteli
subdiagram $\overline{B}$ of $B$. Then for any order on ${B}$, the path space
$X_{\overline{B}}$ has at least one minimal path and one maximal path. Since
$X_{\overline{B}}$ is invariant under $\mathcal{R}$, there are no incoming
edges to vertices of $\overline{B}$ from the vertices that do not lie in
$\overline{B}$. Thus, the maximal and minimal paths which lie in
$X_{\overline{B}}$ stay maximal and minimal also in $X_{B}$. For instance, in
Figure 6, the subdiagram $\overline{B}$ consists of all paths which pass
through the first vertex on each level. More generally, if a generalized
Bratteli diagram $B$ has $n$ subdiagrams with compact $\mathcal{R}$-invariant
path spaces for some $n\in\mathbb{N}$, then any order on $B$ admits at least
$n$ infinite minimal and $n$ infinite maximal paths.

$0$$1$$0$$1$$1$$2$$3$$1$$0$$2$$0$$2$$0$$1$$0$$1$$1$$2$$3$$1$$0$$2$$0$$2$$\ldots$$\ldots$$\ldots$$\vdots$$\vdots$$\vdots$$\vdots$$\vdots$$\vdots$
Figure 6. For the stationary order presented above, the Vershik map is a
continuous bijection, but its inverse is discontinuous. For any order, there
is at least one infinite minimal and one infinite maximal paths.

###  4.2. Prolongation of the Vershik map.

The goal of this subsection is to emphasize a sharp difference in the
properties of Vershik maps for generalized and standard Bratteli diagrams.

######  Theorem 4.7.

There are stationary ordered generalized Bratteli diagrams with a unique
infinite minimal and a unique infinite maximal paths such that

(i) both the Vershik map $\varphi_{B}$ and its inverse $\varphi_{B}^{-1}$ are
not continuous;

(ii) the Vershik map $\varphi_{B}$ is continuous but the inverse
$\varphi_{B}^{-1}$ is discontinuous;

(iii) both the Vershik map $\varphi_{B}$ and its inverse $\varphi_{B}^{-1}$
are continuous.

The proof of this theorem is given in Examples 4.8, 4.10 and Remark 4.9.

We recall that for standard Bratteli diagrams, it is obvious that if a diagram
has a unique minimal infinite path $x_{min}$ and a unique maximal infinite
path $x_{max}$, then the Vershik map which sends $x_{max}$ to $x_{min}$ is a
homeomorphism. This fact follows from compactness of the path space. In the
example below we show that the result does not hold for generalized Bratteli
diagrams.

######  Example 4.8.

The stationary diagram in Figure 7 has a unique infinite minimal path
$x_{min}$ and a unique infinite maximal path $x_{max}$ such that they pass
through the first vertex on each level of the diagram. The second vertex of
the diagram has no minimal or maximal outgoing edges and all maximal and
minimal edges which start not at the first vertex are slanted from right to
left, which guarantees that there are no other infinite minimal and maximal
paths (see also Example 4.2). This means that the corresponding Vershik map
$\varphi_{B}$ sends $x_{max}$ to $x_{min}$ with necessity. We claim that
$\varphi_{B}$ is not a homeomorphism of $X_{B}$ in this case. Indeed, let $x$
be a non-maximal path that coincides with $x_{max}$ for exactly $n$ first
edges, and then passes through a non-maximal edge. Then the image of $x$ under
the Vershik map $\varphi_{B}$ lies in the cylinder set corresponding to the
minimal path of length $n$ which is slanted from right to left and ends in the
second vertex of the diagram. Thus, the images of the non-maximal paths which
lie in the neighborhood of the unique maximal path are not mapped into the
neighborhood of the unique minimal path. Hence the Vershik map $\varphi_{B}$
is not continuous on $X_{B}$. Similarly, $\varphi_{B}^{-1}$ is discontinuous.

$0$$2$$1$$1$$0$$2$$1$$1$$\ldots$$\ldots$$\ldots$$\vdots$$\vdots$$\vdots$$\vdots$$\vdots$$\vdots$$0$$2$$0$$2$$0$$2$$0$$2$$0$$2$$0$$2$$1$$1$$1$$1$
Figure 7. There is a unique infinite minimal and a unique infinite maximal
path, Vershik map $\varphi_{B}$ is a Borel bijection, both $\varphi_{B}$ and
$\varphi_{B}^{-1}$ are discontinuous.

The following remark describes a class of ordered generalized Bratteli
diagrams with a unique infinite minimal path and a unique infinite maximal
path such that the corresponding Vershik map is a homeomorphism.

######  Remark 4.9.

Similar to the case of standard Bratteli diagrams, it is easy to see that if
for every level $n$ of a generalized Bratteli diagram $B$ there is a unique
vertex $v_{min}^{(n)}$ such that all minimal edges of $E_{n}$ start at
$v_{min}^{(n)}$, and a unique vertex $v_{max}^{(n)}$ such that all maximal
edges of $E_{n}$ start at $v_{max}^{(n)}$, then there is a unique infinite
minimal path $x_{min}$, a unique infinite maximal path $x_{max}$, and the
Vershik map which maps $x_{max}$ to $x_{min}$ is a homeomorphism of $X_{B}$.

The example below presents a stationary ordered generalized Bratteli diagram
$B$ with a unique infinite minimal and a unique infinite maximal path such
that the Vershik map $\varphi_{B}$ is a continuous bijection, but
$\varphi^{-1}_{B}$ is discontinuous.

######  Example 4.10.

The stationary ordered diagram on Figure 6 has a unique infinite minimal path
$x_{min}$ and a unique infinite maximal path $x_{max}$ passing through the
first vertex on each level of the diagram. All minimal and maximal edges which
do not end in the first vertex of the diagram are slanted from right to left,
there are no outgoing minimal or maximal edges from the second vertex of the
diagram. Thus, the infinite minimal and infinite maximal paths are unique. We
define the Vershik map $\varphi_{B}$ on these paths by setting
$\varphi_{B}(x_{max})=x_{min}$. Then it is easy to see that $\varphi_{B}$ is a
continuous bijection of $X_{B}$. We show that $\varphi_{B}^{-1}$ is not
continuous. Indeed, any non-maximal path from a neighborhood of $x_{max}$ is
mapped to a neighborhood of $x_{min}$. Now, consider a path $x$ which first
pass through minimal edges and the first vertex of the diagram for $n$ levels,
but then goes once along the minimal edge to the second vertex and then along
the edge $e$ enumerated by $1$ to the third vertex. We see that $x$ lies in a
neighborhood of $x_{min}$, but this path $x$ is not mapped to a neighborhood
of $x_{max}$ by $\varphi^{-1}_{B}$. Indeed, the edge $e$ has a predecessor
which is the edge $e^{\prime}$ labeled by $0$ slanted from right to left. The
source of $e^{\prime}$ is the fourth vertex of the diagram, and it is joined
with $V_{0}$ by a finite maximal path slanted from right to left. Thus, the
preimage of $x$ does not belong to a neighborhood of $x_{max}$.

Recall that some results concerning the prolongation of a Vershik map for
generalized Bratteli diagrams of bounded size can be also found in Section 3.

##  5\. Topological transitivity of the tail equivalence relation

In this section, we prove that the tail equivalence relation on the path space
of a stationary generalized Bratteli diagram with an irreducible and aperiodic
incidence matrix is topologically transitive. We also show that the
irreducibility of the diagram does not imply the tail equivalence relation is
minimal (see Theorem 5.4).

######  Theorem 5.1.

Let $B=(V,E)$ be a generalized stationary Bratteli diagram with an irreducible
aperiodic incidence matrix $F=(f_{ij})_{i,j\in\mathbb{Z}}$. Then the tail
equivalence relation $\mathcal{R}$ is topologically transitive.

###### Proof.

The idea of the proof is to show that the diagram has ‘‘vertical’’ paths with
dense orbits in the path space. We identify vertices at each level with
$\mathbb{Z}$. Fix a vertex $i\in\mathbb{Z}$. Since $F$ is irreducible, there
exists $k\in\mathbb{N}$ such that $f^{(k)}_{ii}>0$. In other words, there is
an infinite path $x$ in $X_{B}$ which passes through the vertex $i$ on levels
$s\cdot k$ for all $s\in\mathbb{N}$. We will show that $x$ is a transitive
point, i.e.,

$\overline{[x]}_{\mathcal{R}}=X_{B}.$

Indeed, it is enough to show that the tail equivalence class of $x$ intersects
every cylinder set. Let $[\overline{e}]=(e_{0},\ldots,e_{N-1})$ be an
arbitrary cylinder set. Denote $j=r(e_{N-1})\in V_{N}$, we show that for
$s\in\mathbb{N}$ large enough there is a path between $i\in V_{sk}$ and $j\in
V_{N}$. Since the matrix $F$ is irreducible, there exists $t\in\mathbb{N}$
such that $f^{(t)}_{ij}>0$. By Lemma A.1, there exists $l=l(j)\in\mathbb{N}$
such that

$f^{(m)}_{jj}>0\mbox{ for all }m\geq l.$

Note that there exist $s\in\mathbb{N}$ and an integer $M\geq l$ such that
there is a path of length $t$ between $i\in V_{sk}$ and $j\in V_{sk-t}$ and a
path of length $M$ between $j\in V_{sk-t}$ and $j\in V_{N}$ (see Figure 8). To
see this, choose $s$ such that $sk-t-N>l$. Take $M=sk-t-N$. Hence, we obtain

$f^{(M+t)}_{ij}\geq f^{(t)}_{ij}\,f^{(M)}_{jj}>0.$

Thus, there exists a path of length $M+t$ between vertex $j=r(e_{N})\in V_{N}$
and vertex $i$ on level $V_{sk}$. Therefore, $[x]_{\mathcal{R}}$ and
$[\overline{e}]$ have a non-empty intersection.

$i\in V_{0}$$j\in V_{N}$$i\in V_{k}$$i\in V_{2k}$$j\in V_{N+M}$$i\in
V_{sk}$$\vdots$ Figure 8. Illustration to the proof of Theorem 5.1. A diagram
with ‘‘vertical’’ paths resulting in dense orbits in the path space.

∎

######  Remark 5.2.

Theorem 5.1 focuses on stationary diagrams. In this remark, we consider a
structural property of the diagram which implies the transitivity of the tail
equivalence relation for non-stationary diagrams as well. Let $B(V,E)$ be an
irreducible generalized Bratteli diagram with a vertex $i\in\mathbb{Z}$
(recall that we identify vertices at each level with integers) such that for
every level $n$ there is an edge between $i\in V_{n}$ and $i\in V_{n+1}$. In
other words, there exists an infinite vertical path $x\in X_{B}$ which passes
through vertex $i$ on each level of the diagram. Then, it can be easily proved
(in the same manner as in the proof of Theorem 5.1) that the orbit
$[x]_{\mathcal{R}}$ is dense in $X_{B}$, and hence the tail equivalence
relation $\mathcal{R}$ is topologically transitive. More generally, it
suffices to assume that there exists a sequence $(n_{k})$ such that there is a
path between the vertex $i\in V_{n_{k}}$ and $i\in V_{n_{k+1}}$ for all $k$.
Then after telescoping, we can use the above result.

It is natural to ask if there exist conditions on the structure of a
generalized Bratteli diagram that would imply the minimality of the tail
equivalence relation. To this effect, we show (see Theorem 5.4) that bounded
size diagrams (see Definition 3.1) with irreducible incidence matrices contain
proper closed subsets that are invariant under the tail equivalence relation
and nowhere dense in the path space. In particular, this shows that the tail
equivalence relation is not minimal. Section 4 provides an example of a
minimal Vershik map on an ordered generalized Bratteli diagram which has no
infinite minimal and no infinite maximal paths (see Example 4.2). Since the
orbits of the Vershik map and the tail equivalence relation are the same for
such a diagram, we obtain an example of a stationary generalized Bratteli
diagram with an irreducible aperiodic incidence matrix such that the
corresponding tail equivalence relation is minimal.

Let $B=(V,E)$ be a generalized Bratteli diagram of bounded size, with the
corresponding sequence $(t_{n},L_{n})_{n\in\mathbb{N}_{0}}$. For $w\in V_{0}$
we define

$Z_{w}^{+}=\left\\{x=(x_{n})\in X_{B}:s(x_{0})\geq w\mbox{ and }r(x_{n})\geq
w+\sum_{i=0}^{n}t_{i}\mbox{ for }n\in\mathbb{N}_{0}\right\\}.$

Similarly, for $w\in V_{0}$ define

$Z_{w}^{-}=\left\\{x=(x_{n})\in X_{B}:s(x_{0})\leq w\mbox{ and }r(x_{n})\leq
w-\sum_{i=0}^{n}t_{i}\mbox{ for }n\in\mathbb{N}_{0}\right\\}.$

######  Lemma 5.3.

Let $B=(V,E)$ be a generalized Bratteli diagram of bounded size, with the
corresponding sequence $(t_{n},L_{n})_{n\in\mathbb{N}_{0}}$. Then, for every
$w\in V_{0}$, the sets $Z_{w}^{+}$, $Z^{-}_{w}$ are invariant with respect to
the tail equivalence relation $\mathcal{R}$.

###### Proof.

Fix $x=(x_{n})\in Z_{w}^{+}$ and consider an infinite path $y=(y_{n})\in
X_{B}$ which is tail equivalent to $x$. Thus there exists $n\in\mathbb{N}$
such that $r(x_{n})=r(y_{n})=v$ for some $v\in V_{n+1}$. Since $x\in
Z_{w}^{+}$, we have

$v\geq w+\sum_{i=0}^{n}t_{i}.$

By Corollary 3.6,

$s(y_{m})\in\left[v-\sum_{i=m}^{n}t_{i},v+\sum_{i=m}^{n}t_{i}\right]\subset\left[w+\sum_{i=1}^{m-1}t_{i},\infty\right)$

for all $m\leq n$. In other words,

$r(y_{m})=s(y_{m+1})\in\left[w+\sum_{i=1}^{m}t_{i},\infty\right).$

Since $x\in Z_{w}^{+}$ and $x$ and $y$ are tail equivalent, we also have

$r(y_{k})\in\left[w+\sum_{i=1}^{k}t_{i},\infty\right)$

for all $k\geq n$. Thus, $y\in Z_{w}^{+}$, this shows that $Z^{+}_{w}$ is
invariant with respect to the tail equivalence relation $\mathcal{R}$. A
similar argument shows that $Z^{-}_{w}$ is also invariant with respect to the
tail equivalence relation $\mathcal{R}$. ∎

We will call the sets $Z^{+}_{w}$, $Z^{-}_{w}$ slanting sets for $w\in V_{0}$.

######  Theorem 5.4.

Let $B=(V,E)$ be a generalized Bratteli diagram of bounded size, with the
corresponding sequence $(t_{n},L_{n})_{n\in\mathbb{N}_{0}}$. Then, for every
$w\in V_{0}$, the sets $Z^{+}_{w}$, $Z^{-}_{w}$ are closed nowhere dense sets
with respect to the topology generated by cylinder sets. In particular, this
shows (together with Lemma 5.3) that the tail equivalence relation
$\mathcal{R}$ is not minimal.

###### Proof.

First, we prove that the set $Z^{+}_{w}$ is closed. Let $Z_{0}$ be a union of
cylinder sets, on level $V_{0}$ which correspond to the vertices in the
interval $[w,\infty)\subset V_{0}$. Let $Z_{n}$ be a union of all cylinder
sets corresponding to finite paths of length $n$ which lie in $Z^{+}_{w}$.
Then we have

$Z_{0}\supset Z_{1}\supset\ldots\supset Z_{n}\supset\ldots,$

each $Z_{n}$ is closed and

$Z^{+}_{w}=\bigcap_{n=0}^{\infty}Z_{n}.$

Hence $Z^{+}_{w}$ is closed.

We show that $Z^{+}_{w}$ does not contain any cylinder set. Let
$\overline{e}=(e_{0},\ldots,e_{n})$ be a finite path which lies in $Z^{+}_{w}$
and $v=r(e_{n})\in V_{n+1}$. There exists $m$ such that
$v-\sum_{i=1}^{m}t_{n+i}<w+\sum_{i=1}^{m}t_{n+i}$. Since
$E(u-t_{n},u)\neq\emptyset$ for all $n$ and for all $u\in V_{n+1}$, there is a
finite path $(e_{n+1},\ldots,e_{n+m})$ between $v\in V_{n+1}$ and
$v-\sum_{i=1}^{m}t_{n+i}\in V_{n+m}$. Hence the cylinder set generated by the
path $(e_{0},\ldots,e_{n+m})$ is a subset of $[\overline{e}]$ which does not
belong to $Z^{+}_{w}$. Thus, $Z^{+}_{w}$ has empty interior. Since $Z^{+}_{w}$
is closed, it follows that $Z^{+}_{w}$ is nowhere dense. A similar argument
shows that $Z^{-}_{w}$ is also closed and nowhere dense. ∎

To end this section, we discuss the cardinality of the sets of the form
$Z^{+}_{w}$ and $Z^{-}_{w}$ for a bounded size generalized Bratteli diagram
and provide conditions that guarantee that $Z^{+}_{w}$ and $Z^{-}_{w}$ are
countable sets.

Let $B=(V,E)$ be a bounded size generalized Bratteli diagram. For every $w\in
V_{0}$, let $Y_{w}^{+}$ denote the set of all infinite paths which start at
$w$ and then pass through the rightmost possible vertex on each level, i.e.
for every $m\in\mathbb{N}$, the paths from $Y_{w}^{+}$ go through the vertex
$w+\sum_{i=0}^{m-1}t_{i}$ on level $m$. Obviously, we have $Y_{w}^{+}\subset
Z_{w}^{+}$ for all $w\in V_{0}$. Analogously, let $Y_{w}^{-}\subset Z_{w}^{-}$
be the set of all infinite paths which start at $w$ and then pass through the
leftmost possible vertex on each level. Note that the sets $Y_{w}^{+}$,
$Y_{w}^{-}$ can be either finite or uncountable (then they are odometers). We
can say that $Y_{w}^{+}$, $Y_{w}^{-}$ are the ‘‘boundary’’ paths for
$Z_{w}^{+}$, $Z_{w}^{-}$.

######  Proposition 5.5.

Let $B=(V,E)$ be a generalized Bratteli diagram of bounded size, with the
corresponding sequence $(t_{n},L_{n})_{n\in\mathbb{N}_{0}}$. If the sets
$Y_{w}^{+}$ are finite for all $w\in V_{0}$ then the sets $Z^{+}_{w}$, are
countable for all $w\in V_{0}$. Otherwise, there exists an uncountable set
$Z^{+}_{w}$. The same is true for $Y_{w}^{-}$ and $Z^{-}_{w}$. In particular,
if $|E(v+t_{n},v)|=|E(v-t_{n},v)|=1$ for all $v\in V_{n+1}$ and
$n\in\mathbb{N}$, then sets $Z^{+}_{w}$, $Z^{-}_{w}$ are countable for all
$w\in V_{0}$.

###### Proof.

Let $Y_{w}^{+}$ be finite for all $w\in V_{0}$. Fix any $w\in V_{0}$. Suppose
$y=(y_{n})_{n=0}^{\infty}\in Z^{+}_{w}$ and $s(y_{0})=u\geq w$. Since $B$ is
of bounded size, for every $m\in\mathbb{N}_{0}$:

$r(y_{m})=u+\widetilde{t}_{0}+\ldots+\widetilde{t}_{m},$

where $\widetilde{t}_{i}\in[-t_{i},t_{i}]$ for $i=1,\ldots,m$. Since $y\in
Z^{+}_{w}$, we have

$r(y_{m})=u+\widetilde{t}_{0}+\ldots+\widetilde{t}_{m}\in\left[w+\sum_{i=0}^{m}t_{i},\infty\right).$

Thus, for all $m\in\mathbb{N}_{0}$,

$u+\sum_{i=0}^{m}\widetilde{t}_{i}\geq w+\sum_{i=0}^{m}t_{i}$

and

$u-w\geq\sum_{i=0}^{m}(t_{i}-\widetilde{t}_{i}).$

Recall that $t_{i}\geq\widetilde{t}_{i}$, hence $t_{i}-\widetilde{t}_{i}\geq
0$. Thus, we have

$u-w\geq\sum_{i=0}^{\infty}(t_{i}-\widetilde{t}_{i}),$

which is possible only if there are finitely many non-zero elements among
$(t_{i}-\widetilde{t}_{i})$. Hence, the path $y$ should go in the same
direction as the ‘‘boundary’’ paths $Y_{w}^{+}$, except for finitely many
deviations. Since all the sets $Y_{w}^{+}$ are finite, the number of such
paths $y$ is countable. If for some $w\in V_{0}$ the set $Y_{w}^{+}$ is
uncountable, then the set $Z_{w}^{+}\supset Y_{w}^{+}$ is also uncountable.
The same proof works for $Y_{w}^{-}$ and $Z_{w}^{-}$. ∎

##  6\. Tail-invariant measures for generalized Bratteli diagrams

In this section, we discuss tail-invariant measures on the path space of a
generalized Bratteli diagram. We emphasize that in this paper, the term
measures is used for non-atomic positive Borel measures. Moreover, we are
mostly interested in full measures, i.e., every cylinder set must be of
positive measure. We consider both finite (probability) and $\sigma$-finite
measures. In the case of $\sigma$-finite measures, we are interested in only
those measures which take finite values on cylinder sets.

We describe every tail-invariant measure in terms of a sequence of positive
vectors associated with vertices of each level, see Theorem 6.6. We give an
explicit construction of ordered generalized Bratteli diagrams for which there
exists no full probability measure invariant under the Vershik map, and the
restriction of the tail equivalence relation onto the equivalence class of
some non-empty clopen set is compressible (see Theorem 6.9). We also provide a
class of generalized Bratteli diagrams such that there exists no tail
invariant (finite or $\sigma$-finite) measure with finite values on cylinder
sets (see Proposition 6.11).

######  Definition 6.1.

Let $B=(V,E)$ be a generalized Bratteli diagram and $\mathcal{R}$ the tail
equivalence relation on the path space $X_{B}$ (see Definition 2.5). A measure
$\mu$ on $X_{B}$ is called tail-invariant if, for any cylinder sets
$[\overline{e}]$ and $[\overline{e}^{\prime}]$ such that
$r(\overline{e})=r(\overline{e}^{\prime})$, we have
$\mu([\overline{e}])=\mu([\overline{e}^{\prime}])$.

######  Remark 6.2.

The theory of countable Borel equivalence relations is a key object in Borel
dynamics; it has been considered from different points of view in numerous
books and articles, see, e.g., [Gao09], [GJ15], [DJK94], [JKL02], [KM04] and
the literature within. For any generalized Bratteli diagram $B$, the tail
equivalence relation $\mathcal{R}$ is a countable Borel hyperfinite
equivalence relation. This means that there exists a Borel automorphism
$T:X_{B}\to X_{B}$ whose orbits coincide with the orbits of $\mathcal{R}$. Can
we take a Vershik map $\varphi_{B}$ for $T$? First, we note that the set of
tail-invariant measures does not depend on an order on $B$. Second, the orbits
of a Vershik map and the tail-invariant relation differ at the sets of maximal
and minimal paths. In general, every measure $\mu$ that is invariant with
respect to a Vershik map is also tail-invariant. Hence, if the sets of maximal
and minimal paths have zero measure, then we can identify tail-invariant
measures with measures invariant with respect to a Vershik map. This happens
for generalized Bratteli diagrams of bounded size, see Lemma 3.8. As a rule,
we will consider measures on the path space of (ordered) Bratteli diagrams
with zero-measure sets of maximal and minimal paths. This property will allow
us to use the notions of tail-invariant measures and that of
$\varphi_{B}$-invariant measures interchangeably, see Section 6.

In what follows we will use the following obvious fact: Suppose that a tail-
invariant Borel measure $\mu$ on $X_{B}$ takes finite values on all cylinder
sets. Then $\mu$ is uniquely determined by its values on cylinder sets in
$X_{B}$, i.e., it can be extended uniquely to all Borel sets.

The definition below uses the notion of Kakutani-Rokhlin towers which is well-
studied in the context of Cantor dynamics. We refer the reader to [HPS92],
[Put18], [GPS95], [BK16] where this notion is discussed.

######  Definition 6.3.

Let $B=(V,E)$ be a generalized Bratteli diagram, for $w\in
V_{n},n\in\mathbb{N}$, denote

$X_{w}^{(n)}=\\{x=(x_{i})\in X_{B}:r(x_{n-1})=w\\}.$

The collection of all such sets forms a partition of $X_{B}$ into Kakutani-
Rokhlin towers corresponding to the vertices from $V_{n}$. Each finite path
$\overline{e}=(e_{0},\ldots,e_{n-1})$ with $r(e_{n-1})=w$, determines a
‘‘floor’’ of this tower

$X_{w}^{(n)}(\overline{e})=\\{x=(x_{i})\in
X_{B}:x_{i}=e_{i},\;i=0,\ldots,n-1\\}$

(we denoted this set by $[\overline{e}]$ above; the notation
$X_{w}^{(n)}(\overline{e})$ indicates the position of $[\overline{e}]$ in the
tower $X_{w}^{(n)}$). Clearly,

$X_{w}^{(n)}=\bigcup_{\overline{e}\in E(V_{0},w)}X_{w}^{(n)}(\overline{e}).$

Thus, the set $X_{w}^{(n)}$ is a union of a finite number of cylinder sets and
can be considered as a tower associated with the vertex $w\in V_{n}$.

######  Definition 6.4.

For $v\in V_{n}$ and $v_{0}\in V_{0}$, we set $h^{(n)}_{v_{0},v}=|E(v_{0},v)|$
and define

$H^{(n)}_{v}=\sum_{v_{0}\in V_{0}}h^{(n)}_{v_{0},v},\ \ n\in\mathbb{N}.$

Set $H^{(0)}_{v}=1$ for all $v\in V_{0}$. This gives us the vector
$H^{(n)}=(H^{(n)}_{v}:v\in V_{n})$ associated with every level
$n\in\mathbb{N}_{0}$. Since $H^{(n)}_{v}=|E(V_{0},v)|$, we call $H^{(n)}_{v}$
the height of the tower $X_{v}^{(n)}$ corresponding to the vertex $v\in
V_{n}$.

######  Remark 6.5.

We have defined the vector $H^{(0)}=(H^{(0)}_{v}:v\in V_{0})$ such that
$H^{(0)}_{v}=1$ for all $v$ (see Definition 6.4). In fact, one can choose any
finite values for $H^{(0)}_{v}$. The role of $H^{(0)}$ can be interpreted as
the vector of heights of the Kakutani-Rokhlin towers between the vertices of
$V_{0}$ and an imaginary level $V_{-1}$ consisting of exactly one vertex.

Observe that

$H^{(n+1)}_{v}=\sum_{w\in V_{n}}f^{(n)}_{vw}H^{(n)}_{w},\ \ \ v\in V_{n+1},$

which immediately implies that

(6.1) $F_{n}H^{(n)}=H^{(n+1)}\ \ \ \textrm{ and }\ \ \ F_{n}\cdots
F_{0}H^{(0)}=H^{(n+1)},\,\,\,n\in\mathbb{N}_{0}.$

We consider here the problem of the existence of tail-invariant measures on
the path space of a generalized Bratteli diagram $B$. Our main results are
mostly related to Bratteli diagrams of bounded size. Note that every incidence
matrix $F_{n}$ defines a linear map from ${\mathbb{R}}^{V_{n}}$ to
${\mathbb{R}}^{V_{n+1}}$ (recall that we identify all $V_{n}$). Using Lemma
3.7, we see that, for every fixed $n\in\mathbb{N}_{0}$ and any
$m\in\mathbb{N}$, we can define the sequence of convex sets

$C^{(n)}_{m}=F_{n}^{T}\cdots F_{n+m-1}^{T}(\mathbb{R}_{+}^{V_{n+m}}),$

where $F_{i}^{T}$ stands for the transpose of $F_{i}$. The above relation is
well defined because $F_{i}^{T}$ maps the positive cone
$\mathbb{R}_{+}^{V_{i+1}}$ into the positive cone of $\mathbb{R}_{+}^{V_{i}}$.
Set

$C^{(n)}_{\infty}=\bigcap_{m=1}^{\infty}C^{(n)}_{m}.$

In general, the set $C^{(n)}_{\infty}$ might be empty.

Given a Bratteli diagram $B$ (generalized or classical), let $\mathcal{M}(B)$
denote the set of tail-invariant finite or $\sigma$-finite measures on the
path space $X_{B}$ which takes finite values on cylinder sets. In the
following theorem, we assume that the set $\mathcal{M}(B)$ is not empty.

######  Theorem 6.6.

Let $B=(V,E)$ be a Bratteli diagram (generalized or classical) with the
sequence of incidence matrices $(F_{n})$. Then:

  1. (1)

If $\mu\in\mathcal{M}(B)$, then for every $n\in\mathbb{N}_{0}$ the vector
defined as follows

(6.2) $p^{(n)}=(\mu(X_{w}^{(n)}(\overline{e})))_{w\in V_{n}}$

satisfies $p^{(n)}\in C_{\infty}^{(n)}$ and

(6.3) $F^{T}_{n}p^{(n+1)}=p^{(n)}$

for all $n\geq 0$.

  2. (2)

Conversely, suppose that $\\{p^{(n)}=(p_{w}^{(n)})\\}_{n\in\mathbb{N}_{0}}$ is
a sequence of non-negative vectors such that $p^{(n)}\in C_{\infty}^{(n)}$ and
$F^{T}_{n}p^{(n+1)}=p^{(n)}$ for all $n\in N_{0}$. Then there exists a
uniquely determined tail-invariant measure $\mu$ such that
$\mu(X_{w}^{(n)}(\overline{e}))=p_{w}^{(n)}$ for $w\in
V_{n},n\in\mathbb{N}_{0}$.

The proof of Theorem 6.6 is straightforward and can be found in [BKMS10] (for
classical Bratteli diagrams) and [BJ21] (for generalized Bratteli diagrams).

######  Remark 6.7.

We stress that part (1) of Theorem 6.6 is true for any generalized Bratteli
diagram whose path space admits a tail-invariant measure. This means also
that, for this diagram, the cone $C_{\infty}^{(n)}$ is not empty for all
$n\in\mathbb{N}_{0}$. In Proposition 6.11, we give an example of a bounded
size diagram such that both sets $\mathcal{M}(B)$ and $C_{\infty}^{(n)}$ are
empty.

Let $\mathcal{E}$ be a countable Borel equivalence relation on a standard
Borel space $X$. It is a well-known fact that the existence of an
$\mathcal{E}$-invariant probability Borel measure $\mu$ on $X$ is determined
by the property of $\mathcal{E}$ called compressibility. For a fixed $x\in X$,
the set $\\{y\in X:(x,y)\in\mathcal{E}\\}$ is called the $\mathcal{E}$-class.
It is said that $\mathcal{E}$ is compressible if there is an injective Borel
map $f:X\to X$ such that for each $\mathcal{E}$-class $L$, $f(L)\subsetneq L$.
A Borel set $A\subset X$ is compressible if the restriction of $\mathcal{E}$
onto $A$ is compressible. We refer to [Nad90], [Nad95] where the following
lemma is proved, (see also [DJK94]).

######  Lemma 6.8.

Let $\mathcal{E}$ be a countable equivalence relation on a standard Borel
space X. The following are equivalent:

  1. (1)

$\mathcal{E}$ is not compressible.

  2. (2)

There is an $\mathcal{E}$-invariant probability measure.

  3. (3)

There is an $\mathcal{E}$-ergodic, $\mathcal{E}$-invariant probability
measure.

Now we give an explicit example of a generalized Bratteli diagram such that
the restriction of the tail equivalence relation onto the equivalence class of
a non-empty clopen set is compressible.

######  Theorem 6.9.

For the one-sided generalized Bratteli diagram $B=(B,\omega)$ with the left-
to-right ordering $\omega$ shown in Fig. 9, we have

  1. (1)

The set $X_{max}$ is empty.

  2. (2)

The Vershik map $\varphi_{B}:X_{B}\to X_{B}\setminus X_{min}$ is a
homeomorphism.

  3. (3)

There exists a non-empty clopen set $C$ such that its tail equivalence class
$\mathcal{R}(C)$ is compressible.

  4. (4)

There is no probability $\varphi_{B}$-invariant measure on $X_{B}$ that
assigns positive values to all cylinder sets.

###### Proof.

The one-sided diagram in Fig. 9 is a modified version of [DK19, Example 7.2].
We identify each vertex level $V_{i}$ with $\mathbb{N}$. Similar to Example
3.9, it is easy to note that there are no infinite maximal paths in the
diagram with respect to the left-to-right order $\omega$. This shows (1). As a
consequence, the Vershik map $\varphi_{B}$ corresponding to $\omega$ is a
homeomorphism from $X_{B}$ to $X_{B}\setminus X_{min}$. Hence we get (2).

To prove (4), we show that there exists a cylinder set $C\subset X_{B}$ such
that $\varphi_{B}(X_{B})=X_{B}\setminus C$. Let $C$ be the cylinder set formed
by all paths that begin at the leftmost vertex of $V_{0}$. The sub-diagram
corresponding to $C$ is a tree and consists only of infinite minimal paths.
Thus, $\varphi_{B}$ maps continuously $X_{B}$ to $X_{B}\setminus C$. The
property

$\varphi_{B}(X_{B})=X_{B}\setminus X_{min}\subset X_{B}\setminus C$

shows that there does not exist any probability $\varphi_{B}$-invariant
measure $\mu$ such that $\mu(C)>0$. Denote by $\mathcal{R}(C)$ the tail
equivalence class of $C$. Since every tail equivalence class $L$ in
$\mathcal{R}(C)$ contains a minimal path, we have $\varphi_{B}(L)\subsetneq L$
for every $L$. Thus, (3) is also proved. ∎

$\vdots$$\vdots$$\vdots$$\vdots$$\vdots$$\vdots$$\vdots$$\vdots$$\ldots$$\ldots$$\ldots$
Figure 9.  A diagram with the left-to-right ordering, and no infinite maximal
paths.

######  Remark 6.10.

Observe that for the classical (standard) Bratteli diagram shown in Fig. 10,
the conclusion (4) of Theorem 6.9 also holds. To see this, we extend the
Vershik map to the entire path space by mapping the unique maximal path to the
unique minimal path in the $2$-odometer (i.e. the subdiagram corresponding to
the vertex $v$ in Fig. 10).

Let $C$ be the cylinder set defined by the edge $[v_{0},u]$. Then it is easy
to see that $\varphi_{B}(X_{B})=X_{B}\setminus C$. Hence, every probability
$\varphi_{B}$-invariant measure $\mu$ must satisfy the condition $\mu(C)=0$
which imply $(4)$. There is a unique probability invariant measure on $X_{B}$
sitting on the minimal component of the tail equivalence relation, the
2-odometer corresponding to the vertex $v$.

$v_{0}$$u$$v$00010011$\vdots$$\vdots$$\vdots$$\vdots$$\vdots$0101010101 Figure
10. Illustration of the result in Theorem 6.9 via a standard Bratteli diagram.

Proposition 6.11, gives an example of a stationary generalized Bratteli
diagram such that there is no tail-invariant measure with finite values on
clopen sets. We emphasize that this example can be generalized to a class of
stationary diagrams with similar property. These diagrams have an incidence
matrix of the form given by (6.4) where the diagonal entries form an
increasing sequence of positive integers.

######  Proposition 6.11.

Let $B=B(F)$ be a one-sided generalized stationary Bratteli diagram as shown
in Figure 11 and given by $\mathbb{N}\times\mathbb{N}$ incidence matrix

(6.4) $F=\begin{pmatrix}2&1&0&0&\ldots\\\ 0&3&1&0&\ldots\\\ 0&0&4&1&\ldots\\\
0&0&0&5&\ldots\\\ \vdots&\vdots&\vdots&\vdots&\ddots\\\ \end{pmatrix}$

There does not exist any tail-invariant measure on $X_{B}$ that assigns finite
values to cylinder sets.

$\ldots$$\ldots$$\ldots$$\vdots$$\vdots$$\vdots$ Figure 11. A Bratteli diagram
with no finite ergodic invariant measure.

###### Proof.

As above, we identify vertices at each level with natural numbers. For
$k\in\mathbb{N}$, denote by $C_{k}$ the cylinder set corresponding to the
vertex $k$ on level $V_{0}$, i.e.

$C_{k}=\\{x\in X_{B}:s(x)=k\\},\ \ \ k\in V_{0}.$

Recall that $s:E\rightarrow V$ is the source map. Suppose that there exists a
non-zero tail-invariant measure $\mu$ on $X_{B}$ which assigns finite values
to cylinder sets. Denote by

$m=\min\\{k\in\mathbb{N}:\mu(C_{k})>0\\}.$

By our assumption, the minimum exists. Normalize the measure $\mu$ such that
$\mu(C_{m})=1$. Let $\overline{e}=(e_{0},\ldots,e_{n})$ be a finite path of
length $(n+1)$ such that the range of $\overline{e}$ is the vertex labeled by
$m$ on level $V_{n}$, i.e., $r(\overline{e})=m\in V_{n}$. As before, we denote
by $[\overline{e}]$ the corresponding cylinder set. It follows from the
definition of $F$ that the path space $X_{B}$ contains countably many
odometers: the set of vertical paths going through a vertex $i$ is an
$(i+1)$-odometer. By tail invariance of $\mu$, all cylinder sets in the
$(m+1)$-odometer have the same measure as the set $[\overline{e}]$ has:

$\mu([\overline{e}])=\frac{1}{(m+1)^{n}}.$

Thus we have

$\mu(C_{m+1})\geq\sum_{n=1}^{\infty}\frac{(m+2)^{n-1}}{(m+1)^{n}}=\infty,$

and this is a contradiction. ∎

######  Remark 6.12.

Let $B(F)$ be a stationary Bratteli diagram as in Proposition 6.11, and let
$\mu$ be any tail-invariant measure. Recall that the vector $p^{(0)}$ (see
(6.2)) consists of the values of the measure $\mu$ of cylinder sets
corresponding to the level $V_{0}$. It follows from the proof of Proposition
6.11 that if, for some vertex $i\in V_{0}$, we have

$0<p_{i}^{(0)}<\infty,$

then $p_{j}^{(0)}=0$ for every $j<i$ and $p_{j}^{(0)}=\infty$ for every $j>i$.
Since the diagram is stationary, the same property holds for every level
$V_{n}$; $n\in\mathbb{N}_{0}$.

##  7\. Uniqueness of tail-invariant measures for stationary Bratteli diagrams

###  7.1. Tail-invariant measures and Perron-Frobenius eigenvectors

In [BJ21], the authors used the Perron-Frobenius (P-F) theory for infinite
matrices to provide a description of tail invariant measures on the path space
of a class of stationary generalized Bratteli diagrams. This class consists of
diagrams with irreducible, aperiodic, and recurrent incidence matrices. for
the reader’s convenience, we provide a brief description of the P-F theory for
infinite matrices in Appendix A. The results formulated in Appendix A are
mostly taken from Chapter $7$ of the book [Kit98]. The foundations of the P-F
theory for infinite matrices were laid down in the 1960s in a series of
articles by D. Vere-Jones [VJ67], [VJ68], [VJ62]. In this section, we use
results from Appendix A, in particular, TheoremA.4 and Theorem A.7 to prove a
criterion for the uniqueness of the tail-invariant measure on the path space
of a stationary generalized Bratteli diagram $B(F)$. Recall that a matrix is
called countably infinite if its rows and columns are indexed by a countable
set. If $F$ is indexed by $\mathbb{N}$, then the diagram $B(F)$ is one-sided
infinite; and if $F$ is indexed by $\mathbb{Z}$, then the diagram is two-sided
infinite. The main results in this section hold for both kinds of diagrams. We
provide examples (see Subsection 7.3) of both kinds of diagrams.

We will keep the following notations: $A$ is the transpose of the infinite
incidence matrix $F$ of the generalized Bratteli diagram $B(F)$. When it
exists (see TheoremA.4), we will denote by $\lambda$ the Perron eigenvalue and
by $\xi=(\xi_{v})$, $\eta=(\eta_{v})$ the right and left eigenvectors for $A$,
i.e., $A\xi=\lambda\xi$ and $\eta A=\lambda\eta$. Note that all entries of
$\xi$ and $\eta$ are positive.

In this section, we will work with an ordered stationary generalized Bratteli
diagram $B(F)=B(V,E,>)$ where $>$ denotes a fixed order. We will assume that
the order $>$ gives rise to a Vershik map on the path space of the diagram. As
before we will denote the corresponding dynamical system by
$(X_{B},\varphi_{B})$. We recall the following result (proved in [BJ22a])
which gives an explicit formula for a tail-invariant measure $\mu$ on the path
space of a stationary generalized Bratteli diagram.

######  Theorem 7.1 (Theorem 2.20, [BJ22a]).

Let $B(F)=B(V,E)$ be a stationary generalized Bratteli diagram such that the
matrix $A=F^{T}$ is infinite, irreducible, aperiodic, and recurrent. Let
$\xi=(\xi_{v})$ be a Perron-Frobenius right eigenvector for $A$, i.e.,
$A\xi=\lambda\xi$, $\xi_{v}>0$.

  1. (1)

There exists a tail-invariant measure $\mu$ on the path space $X_{B}$,
satisfying the following property: if $\overline{e}(w,v)$ is a finite path
that begins at $w\in V_{0}$ and ends at $v\in V_{n}$, $n\in\mathbb{N}$, then

(7.1) $\mu([\overline{e}(w,v)])=\frac{\xi_{v}}{\lambda^{n}},$

where $[\overline{e}(w,v)]$ is the corresponding cylinder set.

  2. (2)

The measure $\mu$ is finite if and only if the Perron eigenvector
$\xi=(\xi_{v})$ has the property $\sum_{v}\xi_{v}<\infty$.

Now we show that if the incidence matrix of a stationary generalized Bratteli
diagram is positive recurrent (in addition to the properties described in
Theorem 7.1), then the invariant measure given by (7.1) is unique. We consider
the cases of finite and $\sigma$-finite measures separately. In the
$\sigma$-finite case (Theorem 7.3), we will work with the assumption that the
dynamical system is conservative. Examples 7.16, 7.9 illustrate Theorems 7.2,
7.3.

######  Theorem 7.2.

Let $B(F)=B(V,E,>)$ be an ordered stationary generalized Bratteli diagram such
that the matrix $A=F^{T}$ is infinite, irreducible, aperiodic, and positive
recurrent. Let $\xi=(\xi_{v})$ be a Perron-Frobenius right eigenvector for $A$
such that $\sum_{u\in V_{0}}\xi_{u}=1$. Then the measure $\mu$ given in (7.1)
is the unique probability $\varphi_{B}$-invariant measure that takes positive
values on cylinder sets.

###### Proof.

We fix a vertex $w\in V_{n}$, and consider a cylinder set given by
$[\overline{e}]=(e_{0},\ldots,e_{n-1})$ with $r(e_{n-1})=w$. Then by (7.1), we
have

$\mu([\overline{e}])=\frac{\xi_{w}}{\lambda^{n}}.$

As proved in Theorem 7.1, the measure $\mu$ is probability and takes finite
positive values on cylinder sets. Suppose that $\nu$ is a probability ergodic
$\varphi_{B}$-invariant measure with positive values on cylinder sets. Let
$N\geq n$ and $v\in V_{N}$ be such that $E(w,v)\neq\emptyset$. Then, by the
Birkhoff ergodic theorem,

$\nu([\overline{e}])=\lim_{N\rightarrow\infty}\frac{|E(w,v)|}{H_{v}^{(N)}},$

where $H_{v}^{(N)}$ is the total number of finite paths with the range at
vertex $v\in V_{N}$ (see Definition 6.4). Since $A$ is positive recurrent, it
follows from Theorem A.7 that, for any $v,w\in V$,

$\underset{N\rightarrow\infty}{\mathrm{lim}}\,\,\dfrac{a_{wv}^{(N)}}{\lambda^{N}}=\xi_{w}\eta_{v},$

where $a_{wv}^{(N)}$ is the entry of $A^{N}$ and $\eta=(\eta_{v})$ is the left
eigenvector of $A$ normalized by the condition $\eta\cdot\xi=1$. Thus, we
obtain

$\nu([\overline{e}])=\lim_{N\rightarrow\infty}\frac{a^{(N-n)}_{wv}}{\sum_{u\in
V_{0}}a^{(N)}_{uv}}=\frac{\xi_{w}\eta_{v}}{\lambda^{n}\sum_{u\in
V_{0}}\xi_{u}\eta_{v}}=\frac{\xi_{w}}{\lambda^{n}}=\mu([\overline{e}]).$

∎

Now we show the uniqueness of the infinite $\sigma$-finite
$\varphi_{B}$-invariant measure given by (7.1). We work with an additional
assumption that the dynamical system $(X_{B},\varphi_{B},\mu)$ is
conservative.

######  Theorem 7.3.

Let $B(F)=B(V,E,>)$ be an ordered stationary generalized Bratteli diagram such
that the matrix $A=F^{T}$ is infinite, irreducible, aperiodic, and positive
recurrent. Let $\xi=(\xi_{v})$ be a Perron-Frobenius right eigenvector for $A$
such that $\sum_{u\in V_{0}}\xi_{u}=\infty$. Let $\mu$ be the $\sigma$-finite
$\varphi_{B}$-invariant (given by (7.1)) such that $(X_{B},\varphi_{B},\mu)$
is conservative. Then $\mu$ is the unique (up to a constant multiple)
$\sigma$-finite $\varphi_{B}$-invariant ergodic measure that takes positive
values on cylinder sets.

###### Proof.

By Theorem 7.1, there exist an invariant $\sigma$-finite measure $\mu$ on the
path space of a generalized Bratteli diagram with irreducible, aperiodic and
recurrent incidence matrix. Consider two cylinder sets
$[\overline{e_{1}}],\,[\overline{e_{2}}]\subset X_{B}$ such that
$r(\overline{e_{1}})=v_{1}\in V_{n_{1}}$ and $r(\overline{e_{2}})=v_{2}\in
V_{n_{2}}$. Without loss of generality, assume that $n_{2}>n_{1}$. Using
(7.1), we calculate the ratio of their measures,

(7.2)
$\dfrac{\mu([\overline{e_{1}}])}{\mu([\overline{e_{2}}])}=\dfrac{\xi_{v_{1}}/\lambda^{n_{1}}}{\xi_{v_{2}}/\lambda^{n_{2}}}=\dfrac{\xi_{v_{1}}}{\xi_{v_{2}}}\lambda^{(n_{2}-n_{1})}.$

Let $m$ be a $\sigma$-finite ergodic measure on $X_{B}$. Now we apply Hopf’s
ratio ergodic theorem for $m$ and find the ratio of measures of the same
cylinder sets $[\overline{e_{1}}]$ and $[\overline{e_{2}}]$ (see [Aar97] for
references). For this, let $N>n_{2}$ and $w\in V_{N}$ be such that the sets
$E(v_{1},w)$ and $E(v_{2},w)$ are non empty. Since $A$ is irreducible, we can
choose such $N$ using Lemma A.1(i). Now, we apply Hopf’s ratio ergodic theorem
to obtain

$\dfrac{m([\overline{e_{1}}])}{m([\overline{e_{2}}])}=\underset{N\rightarrow\infty}{\mathrm{lim}}\,\,\dfrac{|E(v_{1},w)|}{|E(v_{2},w)|}\,\,\,=\underset{N\rightarrow\infty}{\mathrm{lim}}\,\,\dfrac{a_{v_{1}w}^{(N-n_{1})}}{a_{v_{2}w}^{(N-n_{2})}}=\underset{N\rightarrow\infty}{\mathrm{lim}}\,\,\dfrac{a_{v_{1}w}^{(N-n_{1})}}{\lambda^{N-n_{1}}}\cdot\dfrac{\lambda^{N-n_{2}}}{a_{v_{2}w}^{(N-n_{2})}}\cdot\lambda^{n_{2}-n_{1}}.$

Since $A$ is positive recurrent, it follows from Theorem A.7 that, for any
$v,w\in V$,

$\underset{N\rightarrow\infty}{\mathrm{lim}}\,\,\dfrac{a_{vw}^{(N)}}{\lambda^{N}}=\xi_{v}\eta_{w}.$

Therefore,

(7.3)
$\dfrac{m([\overline{e_{1}}])}{m([\overline{e_{2}}])}=\dfrac{\xi_{v_{1}}\,\,\eta_{w}}{\xi_{v_{2}}\,\,\eta_{w}}\lambda^{n_{2}-n_{1}}=\dfrac{\xi_{v_{1}}}{\xi_{v_{2}}}\lambda^{n_{2}-n_{1}}=\dfrac{\mu([\overline{e_{1}}])}{\mu([\overline{e_{2}}])}.$

Relation (7.3) shows that, for every cylinder set $[\overline{e}]$, the ratio

$\frac{m([\overline{e}])}{\mu([\overline{e}])}=c$

for some constant $c$. This means that $m=c\mu$, and the proof is complete. ∎

###  7.2. Generalized Bratteli diagrams with finite tail invariant measures

Suppose that a matrix $A=(a_{ij})$ with $a_{ij}\in\mathbb{N}_{0}$ for all
$i,j$ is such that the Perron-Frobenius theorem holds: there exists a finite
Perron eigenvalue $\lambda$ and a non-negative right eigenvector
$\xi=(\xi_{i})$ such that $A\xi=\lambda\xi.$ We give sufficient conditions on
the matrix $A$ that lead to the existence of a summable eigenvector $\xi$,
i.e., $\sum_{i}\xi_{i}<\infty$. By Theorem 7.1, these conditions will
guarantee the existence of a finite tail-invariant measure.

######  Proposition 7.4.

Let $A=(a_{i,j}:i,j\in\mathbb{N})$ be a non-negative matrix such that there
exists a positive eigenvector $\xi$ corresponding an eigenvalue $\lambda$. If
there exists a row of $A$ with finitely many zero entries, then the
eigenvector $\xi=(\xi_{i})$ is summable:

$\sum_{i\in\mathbb{N}}\xi_{i}<\infty.$

###### Proof.

Let the $i$th row of $A$ have finitely many zero entries:

$a_{ij}=0\ \Longleftrightarrow\ (j\in I,\;|I|<\infty).$

From the equality

$\sum_{j\in\mathbb{N}}a_{ij}\xi_{j}=\lambda\xi_{i}$

we have

$\sum_{j\in\mathbb{N}}\xi_{j}=\sum_{j\in I}\xi_{j}+\sum_{j\notin
I}\xi_{j}\leq\sum_{j\in
I}\xi_{j}+\sum_{j\in\mathbb{N}}a_{ij}\xi_{j}=\sum_{j\in
I}\xi_{j}+\lambda\xi_{i}<\infty.$

∎

Note that the matrix $A$ will also have a summable right eigenvector if there
are two rows in $A$ such that one row contains non-zero elements in even-
numbered columns and the other one in odd-numbered columns. More generally,
the following result holds.

######  Proposition 7.5.

Let $A$ be a $\mathbb{N}\times\mathbb{N}$ matrix with non-negative integer
entries such that $A\xi=\lambda\xi$ for $0<\lambda<\infty$ and $\xi>0$. For
every $k\in\mathbb{N}$, let

$M_{k}=\\{j\in\mathbb{N}:a_{kj}>0\\}.$

Assume that there exists a finite collection of rows
$\\{k_{1},\ldots,k_{p}\\}$ such that

$|\mathbb{N}\setminus\bigcup_{t=1}^{p}M_{k_{p}}|<\infty.$

Then the eigenvector $\xi=(\xi_{i})$ is finite in the sense that

$\sum_{i\in\mathbb{N}}\xi_{i}<\infty.$

###### Proof.

For each $t=1,\ldots,p$, we have

$\sum_{j\in\mathbb{N}}a_{k_{t}j}\xi_{j}=\sum_{j\in
M_{k_{t}}}a_{k_{t}j}\xi_{j}=\lambda\xi_{k_{t}}.$

Consider the sum of these relations:

$\sum_{t=1}^{p}\sum_{j\in
M_{k_{t}}}a_{k_{t}j}\xi_{j}=\lambda\sum_{t=1}^{p}\xi_{k_{t}}.$

Note that since $a_{k_{t}j}\geq 1$ for $j\in M_{k_{t}}$, we have

$\sum_{t=1}^{p}\sum_{j\in
M_{k_{t}}}a_{k_{t}j}\xi_{j}\geq\sum_{j\in\bigcup_{t=1}^{p}M_{k_{p}}}\xi_{j}.$

Then we get

$\displaystyle\sum_{j\in\mathbb{N}}\xi_{j}=$
$\displaystyle\sum_{j\in\mathbb{N}\setminus\bigcup_{t=1}^{p}M_{k_{p}}}\xi_{j}+\sum_{j\in\bigcup_{t=1}^{p}M_{k_{p}}}\xi_{j}$
$\displaystyle\leq$
$\displaystyle\sum_{j\in\mathbb{N}\setminus\bigcup_{t=1}^{p}M_{k_{p}}}\xi_{j}+\lambda\sum_{t=1}^{p}\xi_{k_{t}}<\infty,$

since the set $\mathbb{N}\setminus\bigcup_{t=1}^{p}M_{k_{p}}$ is finite. ∎

######  Remark 7.6.

The converse of Proposition 7.5 is not true since there are banded matrices
with probability right eigenvectors, see examples in Subsection 7.3.

###  7.3. Examples

In this subsection we consider several classes of stationary generalized
Bratteli diagrams that admit finite and $\sigma$-finite tail-invariant
measures on their path spaces. The reader can find the proof of these results
in Appendix B.

######  Example 7.7.

For $a,b\in\mathbb{N}$, consider the generalized stationary Bratteli diagram
$B(F_{1})$ where $A_{1}=F_{1}^{T}$ is given by

(7.4)
$A_{1}=\left(\begin{array}[]{cccccccccccc}\ddots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\udots\\\
\cdots&2b&0&a&0&0&\textbf{0}&0&0&0&0&\cdots\\\
\cdots&0&2b&0&a&0&\textbf{0}&0&0&0&0&\cdots\\\
\cdots&0&0&2b&0&b&\textbf{0}&0&0&0&0&\cdots\\\
\cdots&0&0&0&2b&a&\textbf{b}&0&0&0&0&\cdots\\\
\cdots&\textbf{0}&\textbf{0}&\textbf{0}&\textbf{0}&\textbf{b}&\textbf{a}&\textbf{2b}&\textbf{0}&\textbf{0}&\textbf{0}&\cdots\\\
\cdots&0&0&0&0&0&\textbf{b}&0&2b&0&0&\cdots\\\
\cdots&0&0&0&0&0&\textbf{0}&a&0&2b&0&\cdots\\\
\cdots&0&0&0&0&0&\textbf{0}&0&a&0&2b&\cdots\\\
\udots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\ddots\\\
\end{array}\right)$

We use the bold font to indicate the 0-th row and 0-th column. Remark that
this matrix is also considered in Example (8.4) from a different point of
view. In [BJS22], it was shown that for $a=b=1$, one can model by $B(F_{1})$
endowed with the left-to-right order a substitution dynamical system given by
the so-called ‘‘one step forward, two steps back’’ substitution on
$\mathbb{Z}$:

$\displaystyle-1\mapsto-2\;-1\;0;\quad 0\mapsto-1\;0\;1;$ $\displaystyle
n\mapsto(n-1)(n+1)(n+1)\mbox{ for }n\leq-2;$ $\displaystyle
n\mapsto(n-1)(n-1)(n+1)\mbox{ for }n\geq 1.$

######  Proposition 7.8.

The stationary generalized Bratteli diagram $B(F_{1})$, where
$A_{1}=F_{1}^{T}$ as in (7.4), supports a tail-invariant measure $\mu$ given
by (7.1). The measure $\mu$ is defined using the eigenvalue $\lambda=a+2b$ and
the corresponding right eigenvector $\xi$ of $A_{1}$ given by

$\xi=\left(\ldots,\dfrac{1}{2^{3}}\left(\dfrac{a}{b}\right)^{2},\dfrac{1}{2^{2}}\left(\dfrac{a}{b}\right),\dfrac{1}{2},1,\mathbf{1},\dfrac{1}{2},\dfrac{1}{2^{2}}\left(\dfrac{a}{b}\right),\dfrac{1}{2^{3}}\left(\dfrac{a}{b}\right)^{2},\ldots\right)^{T}.$

(The 0-th entry of $\xi$ is shown in bold font). Moreover, $\mu$ is finite if
and only if $a<2b$.

We prove this result in Appendix B.

######  Example 7.9.

(see also [BB16]) This example deals with a Bratteli diagram of different
nature. This diagram is defined by a null-recurrent matrix of period $2$ and
has at least two infinite $\sigma$-finite measures which take finite positive
values on cylinder sets.

Let $a$ and $b$ be natural numbers. Consider the generalized stationary
Bratteli diagram $B(F_{2})$ where $A_{2}=F_{2}^{T}$ is given by

(7.5)
$A_{2}=A_{2}(a,b)=\left(\begin{array}[]{cccccccccccc}\ddots&\vdots&\vdots&\vdots&\vdots&\vdots&\udots\\\
\cdots&0&{b}&0&0&0&\cdots\\\ \cdots&{a}&0&{b}&0&0&\cdots\\\
\cdots&0&{a}&0&{b}&0&\cdots\\\ \cdots&0&0&{a}&{0}&{b}&\cdots\\\
\cdots&0&0&0&{a}&{0}&\cdots\\\
\udots&\vdots&\vdots&\vdots&\vdots&\vdots&\ddots\\\ \end{array}\right)$

######  Proposition 7.10.

(1) If $a\neq b$, then there exist at least two tail-invariant infinite
$\sigma$-finite measures on the path space $X_{B}$, $\mu$ and
$\widetilde{\mu}$. According to (7.1), the measure $\mu$ is determined by
$\lambda=a+b$ and $\xi=\left(\ldots,1,1,1\ldots\right)^{T}$, and the measure
$\widetilde{\mu}$ is determined by $\widetilde{\lambda}=2\sqrt{ab}$ and
$\widetilde{\xi}=\left(\widetilde{\xi}_{n}\right)^{T}$ where
$\widetilde{\xi}_{n}=\left(\dfrac{a}{b}\right)^{\tfrac{n}{2}}$,
$n\in\mathbb{Z}$.
(2) If $a=b$, then $\mu=\widetilde{\mu}$.
The matrix $A_{2}(a,b)$ is null recurrent for any $a,b\in\mathbb{N}$.

This matrix is also considered in Example 8.2 where we use the corresponding
stochastic matrix to show that $A_{2}(a,b)$ is null recurrent for any
$a,b\in\mathbb{N}$. We prove Proposition 7.10 in Appendix B.

Examples of one-sided infinite Bratteli diagrams. In this subsection, we
consider examples of one-sided infinite stationary generalized Bratteli
diagrams and find conditions under which infinite tri-diagonal matrices
(indexed by $\mathbb{N}$) have eigenvectors with finite entry sum (hence admit
finite invariant measure).

Let $A=(a_{ij})_{i,j\in\mathbb{N}}$ be a one-sided infinite tri-diagonal
matrix:

$A=\begin{pmatrix}a_{11}&a_{12}&0&0&0&...\\\ a_{21}&a_{22}&a_{23}&0&0&...\\\
0&a_{32}&a_{33}&a_{34}&0&...\\\ 0&0&a_{43}&a_{44}&a_{45}&...\\\
0&0&0&a_{54}&a_{55}&...\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\ddots\end{pmatrix}$

We assume that $A$ has an eigenvalue $\lambda$ with the right eigenvector
$\xi$, i.e. $A\xi=\lambda\xi$. Our goal is to find conditions when the vector
$\xi$ is summable, $\sum_{i=1}^{\infty}\xi_{i}<\infty$. We denote by

$\sigma_{i}=\sum_{j=1}^{\infty}a_{ij}$

the sum of the $i$-th row of $A$. It turns out that an important class of
examples comes from the matrices having equal column sum property together
with an additional requirement on the row sum as described in the definition
below.

######  Definition 7.11.

We say that a one-sided infinite matrix $A=(a_{ij})_{i,j\in\mathbb{N}}$ is
balanced if it satisfies following conditions :

  1. (1)

$A$ has the property of equal column sum, i.e. $\sum_{i=1}^{\infty}a_{ij}=c$
for every $j\in\mathbb{N}$; this automatically implies that $c=\lambda$ for
every $j\in\mathbb{N}$.

  2. (2)

$\sigma_{2}=\sigma_{3}=\sigma_{4}=....$ and $\sigma_{1}>\sigma_{i}$ for all
$i>1$.

Below we provide some examples of one-sided infinite stationary Bratteli
diagrams with balanced incidence matrices.

######  Example 7.12.

Fix $b,c,\alpha\in\mathbb{N}$ such that $\alpha>1$. Consider the generalized
diagram $B(F_{3})$ where $A_{3}=F_{3}^{T}$ is given by

(7.6) $A_{3}=\begin{pmatrix}b+\alpha c&\alpha c&0&0&0&...\\\ c&b&\alpha
c&0&0&...\\\ 0&c&b&\alpha c&0&...\\\ 0&0&c&b&\alpha c&...\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\ddots\end{pmatrix}$

######  Proposition 7.13.

The stationary generalized Bratteli diagram $B(F_{3})$, where
$A_{3}=F_{3}^{T}$ is defined in (7.6), supports a tail-invariant measure $\mu$
defined by (7.1) using the eigenvalue $\lambda=b+c+\alpha c$ and the
corresponding right eigenvector $\xi=(\xi_{n})_{n\in\mathbb{N}}$ for $A_{3}$
such that

$\xi=\Big{(}\xi_{1},\dfrac{\xi_{1}}{\alpha},\dfrac{\xi_{1}}{\alpha^{2}},\cdots,\dfrac{\xi_{1}}{\alpha^{n-1}},\cdots\Big{)}^{T}$

where $\xi_{1}$ can be chosen to be any positive integer.

Proposition 7.13 is proved in Appendix B.

######  Example 7.14.

Fix $b\in\mathbb{N}_{0}$, $r\in\mathbb{N}$ and take integers $\alpha$ and
$\beta$ such that $|\alpha|,|\beta|<r$. Denote by

(7.7) $q_{1}=\dfrac{r-\alpha}{r+\beta},\qquad
q_{2}=\dfrac{r-\beta}{r+\alpha}.$

Consider the one-sided infinite stationary Bratteli diagram $B(F_{4})$ where
$A_{4}=F_{4}^{T}$ is given by

(7.8) $A_{4}=F_{4}^{T}=\begin{pmatrix}(b+r+\alpha)&r+\beta&0&0&0&...\\\
r-\alpha&b&r+\alpha&0&0&...\\\ 0&r-\beta&b&r+\beta&0&...\\\
0&0&r-\beta&b&r+\beta&...\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\ddots\end{pmatrix}$

######  Proposition 7.15.

For the stationary Bratteli diagram $B(F_{4})$ defined by the matrix given in
(7.4), the following statements hold.

  1. (1)

For the eigenvalue $\lambda=b+2r$, the right eigenvector $\xi=(\xi_{n})$
($A\xi=\lambda\xi$) for $A_{4}$ has the entries

(7.9)
$x_{1}=1,\qquad\xi_{2n}=\dfrac{(r-\alpha)^{n}\,(r-\beta)^{n-1}}{(r+\alpha)^{n-1}(r+\beta)^{n}},\qquad\xi_{2n+1}=\dfrac{(r-\alpha)^{n}(r-\beta)^{n}}{(r+\alpha)^{n}(r+\beta)^{n}};$

  2. (2)

If $r,\alpha,\beta$ are chosen such that $q_{1}q_{2}<1$ (see (7.7)), then the
diagram $B(F_{4})$ supports a finite tail invariant measure $\mu$ determined
as in (7.1).

  3. (3)

If $r,\alpha,\beta$ are chosen such that $q_{1}q_{2}\geq 1$, the tail-
invariant measure is $\sigma$-finite.

The proof of Proposition 7.15 is given in Appendix B.

We finish this section with two examples of generalized Bratteli diagrams of
different types: they are not of bounded size.

######  Example 7.16 (Renewal subshift).

Consider the one-sided infinite generalized diagram $B(F_{5})$ where
$A_{5}=F_{5}^{T}=(a_{ij})_{i,j\in\mathbb{N}}$ is defined by

(7.10) $A_{5}=\begin{pmatrix}1&1&1&1&1&1&\cdots\\\ 1&0&0&0&0&0&\cdots\\\
0&1&0&0&0&0&\cdots\\\ 0&0&1&0&0&0&\cdots\\\ 0&0&0&1&0&0&\cdots\\\
0&0&0&0&1&0&\cdots\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\ddots\end{pmatrix}.$

In other words,

(7.11) $a_{ij}=\left\\{\begin{aligned} &1\mbox{ if }i=1\mbox{ or }i=j+1,\\\
&0\mbox{ otherwise }.\end{aligned}\right.$

We remark that the matrix $A_{5}$ is balanced, see Definition 7.11.

######  Proposition 7.17.

The stationary generalized Bratteli diagram $B(F_{5})$ corresponding to the
renewal shift in Example 7.16 supports a unique probability ergodic tail
invariant measure $\mu$. The measure $\mu$ is defined by (7.1) using the
Perron eigenvalue $\lambda=2$ and the corresponding right eigenvector
$\xi=\left(\frac{1}{2},\frac{1}{2^{2}},\frac{1}{2^{3}},\ldots\right)^{T}$ of
$A_{5}$. The matrix $A_{5}$ is positive recurrent.

The proof follows from an application of Theorem 7.2 and is given in Appendix
B.

Rest of the examples in this section consist of generalized Bratteli diagrams
with incidence matrices that are not balanced.

######  Example 7.18 (Pair renewal shift, see [Ras21]).

Consider the one-sided infinite generalized diagram $B(F_{6})$ where
$A_{6}=F_{6}^{T}=(a_{ij})_{i,j\in\mathbb{N}}$ is given by

(7.12) $a_{ij}=\left\\{\begin{aligned} &1,\mbox{ if }i=1\mbox{ and
}j\in\mathbb{N},\\\ &1,\mbox{ if }i=2\mbox{ and }j=2n\mbox{ for
}n\in\mathbb{N},\\\ &1,\mbox{ if }i=n+1\mbox{ and }j=n\mbox{ for
}n\in\mathbb{N},\\\ &0\mbox{ otherwise}.\end{aligned}\right.$

Note that $A_{6}$ does not have the equal column sum property and is not
balanced. Explicitly, the matrix $A_{6}$ is

$A_{6}=\begin{pmatrix}1&1&1&1&1&1&\cdots\\\ 1&1&0&1&0&1&\cdots\\\
0&1&0&0&0&0&\cdots\\\ 0&0&1&0&0&0&\cdots\\\ 0&0&0&1&0&0&\cdots\\\
0&0&0&0&1&0&\cdots\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\ddots\end{pmatrix}.$

This infinite matrix is a modified version of $A_{5}$, and the corresponding
shift space is called the pair renewal shift (see [Ras21]).

######  Proposition 7.19.

The stationary generalized Bratteli diagram $B(F_{6})$ corresponding to the
pair renewal shift in Example 7.18 supports a unique up constant multiple
finite tail-invariant measure $\mu$ given by (7.1). The measure $\mu$ is
defined by the Perron eigenvalue $\lambda=1+\sqrt{2}$ and the corresponding
right eigenvector $\xi=(\xi_{n})_{n\in\mathbb{N}}$ of $A_{6}$ which is given
by

$\xi_{1}=\frac{1}{1+\sqrt{2}},\qquad\xi_{n}=\frac{2}{(1+\sqrt{2})^{n}},\quad
n\geq 2.$

The matrix $A_{6}$ is positive recurrent.

The proof of Proposition 7.19 is given in Appendix B.

######  Example 7.20.

Consider the one-sided infinite generalized diagram $B(F_{7})$ where
$A_{7}=F_{7}^{T}=(a_{ij})_{i,j\in\mathbb{N}}$ is defined by

(7.13) $A_{7}=\begin{pmatrix}c_{0}&1&0&0&0&0&\cdots\\\
c_{1}&0&1&0&0&0&\cdots\\\ c_{2}&0&0&1&0&0&\cdots\\\ c_{3}&0&0&0&1&0&\cdots\\\
c_{4}&0&0&0&0&1&\cdots\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\ddots\end{pmatrix}$

Here $c_{k}\in\mathbb{N}$ for all $k\geq 0$.

######  Proposition 7.21.

The stationary generalized Bratteli diagram $B(F_{7})$ corresponding to the
matrix defined in (7.13) supports a $\sigma$-finite infinite tail-invariant
measure $\mu$ given by (7.1) if there exists constant $C\in\mathbb{N}$ such
that for every $k\in\mathbb{N}_{0}$, $c_{k}<C$. The measure $\mu$ is defined
by the eigenvalue $\lambda\leq C+1$ and the corresponding right eigenvector