#  High-$\mathbf{Q^{2}}$ Behavior of Structure Function $\mathbf{F_{2}}$ with
the Solution to Balitsky-Kovchegov Equation

Wei Kou  Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou
730000, China  University of Chinese Academy of Sciences, Beijing 100049,
China  Gang Xie  Institute of Modern Physics, Chinese Academy of Sciences,
Lanzhou 730000, China  University of Chinese Academy of Sciences, Beijing
100049, China  Guangdong Provincial Key Laboratory of Nuclear Science,
Institute of Quantum Matter, South China Normal University, Guangzhou 510006,
China  Xiaopeng Wang  Institute of Modern Physics, Chinese Academy of
Sciences, Lanzhou 730000, China  University of Chinese Academy of Sciences,
Beijing 100049, China  Lanzhou University, Lanzhou 730000, China  Chengdong
Han<EMAIL_ADDRESS>Institute of Modern Physics, Chinese Academy of
Sciences, Lanzhou 730000, China  University of Chinese Academy of Sciences,
Beijing 100049, China  Xurong Chen<EMAIL_ADDRESS>Institute of Modern
Physics, Chinese Academy of Sciences, Lanzhou 730000, China  University of
Chinese Academy of Sciences, Beijing 100049, China  Guangdong Provincial Key
Laboratory of Nuclear Science, Institute of Quantum Matter, South China Normal
University, Guangzhou 510006, China

###### Abstract

As a lot of studies have declared, the proton structure function $F_{2}$
explicitly satisfies a power-law mathematical formation. We describe the
behavior of proton structure function $F_{2}$ at small-$x$ as well as large
$Q^{2}$ using the analytical solution of the Balitsky-Kovchegov equation. We
obtain the effective slope value $\lambda$ of the structure function with
$Q^{2}$. Our results come from fitting the HERA experimental data at low
$Q^{2}$ region and we also give the prediction for the effective slope of
$F_{2}$ at the high $Q^{2}$ region.

###### pacs:

24.85.+p, 13.60.Hb, 13.85.Qk

##  I Introduction

Quantum chromodynamics (QCD) is used to explain the interactions between
hadrons, especially the high-energy scattering processes. However, due to the
color confinement nature of QCD, we cannot obtain pure interaction between
quarks and gluons, which are crucial for high-energy physics. Fortunately, due
to the asymptotic freedom of QCD Gross and Wilczek (1973); Politzer (1973),
the behavior in the high energy region can be solved using perturbation
theory. Experimentally, the high-energy lepton-nucleon scattering experiments
of recent years have given physicists a window to study the strong
interactions. Lots of hadron structure-related issues need to be considered.

In last two decades, the LO and NLO of the Balitsky-Fadin-Kuraev-Lipatov
(BFKL) equation Lipatov (1976); Kuraev  _et al._ (1977); Balitsky and Lipatov
(1978) were studied by L. N. Lipatov et al. Ellis  _et al._ (2008); Kowalski
_et al._ (2010, 2011, 2013, 2014, 2016, 2017); Salam (1998), using the Green’s
function approach. They argued that the BFKL equation shall be considered as
an eigen-equation with eigenvalue $\omega_{n}$. The authors in Ref. Ellis  _et
al._ (2008) argued that the discrete asymptotically-free BFKL Pomeron has been
shown firstly to describe the HERA data at low $x$ and high $Q^{2}$. Most of
the Deep inelastic scattering (DIS) data provided by HERA are well described
by the Dokshitzer–Gribov–Lipatov–Altarelli–Parisi (DGLAP) equations Dokshitzer
(1977); Gribov and Lipatov (1972); Lipatov (1974); Altarelli and Parisi (1977)
equation presented by the renormalization group evolution Wilson (1975).
However, the discussion related to HEAR data for small $x$ regions was
initiated with the combination of the DGLAP and BFKL equations. In other
words, since the variation of the scattering cross section with energy has to
satisfy the unitarity limit naturally, the BFKL as a linear evolution equation
will destroy this property in the high energy region. Fortunately, the
infinite growth of gluons seems to be suppressed by certain processes. The
saturation behavior of gluons is discussed in Ref. Mueller (2001); Stasto  _et
al._ (2001); Munier and Peschanski (2003) (and references cited therein).
Balitsky-Kovchegov (BK) equation Balitsky (1997); Kovchegov (1999, 2000);
Balitsky (2001), a nonlinear equation representing the evolution of gluons,
which is a mean-field approximation to the Jalilian-Marian-Iancu-McLerran-
Weigert-Leonidov-Kovner (JIMWLK) equation Balitsky (1996); Jalilian-Marian
_et al._ (1997); Iancu  _et al._ (2001); Weigert (2002). The BK equation
describes the gluon behavior in the small $x$ region. Some of us obtained the
analytical solution of the BK equation at momentum space in a novel way in
previous work Wang  _et al._ (2021).

HERA collaboration provides high precision data for the proton structure
function $F_{2}$, therefore, it is used to study the behavior of $F_{2}$ at
the small $x$ and large $Q^{2}$ conditions. The behavior of the structure
function at small $x$ region is generally understood by studying its slope
parameter $\lambda$ Bartels and Kowalski (2001), with the form
$\lambda\propto\partial\ln F_{2}/\partial\ln(1/x)$. A significant amount of
previous works include this topic in a comprehensive manner Ellis  _et al._
(2008); Kowalski  _et al._ (2010, 2011, 2013, 2014, 2016, 2017); Salam (1998);
Cvetic  _et al._ (2009); Cooper-Sarkar  _et al._ (1998); Kotikov and Parente
(1999); Kotikov (2007); Ball and Forte (1994); Illarionov  _et al._ (2008);
Mankiewicz  _et al._ (1997) (and references cited therein). Until recent
years, physicists also given different explanations and fitting results with
new HERA data Abramowicz  _et al._ (2015); Luszczak and Kowalski (2020). In
contrast to the range of $Q^{2}$ included in the data, several theoretical
works give results for the parameter lambda versus $Q^{2}$ for only the
intermediate region such as $\mathcal{O}(200)$ MeV2 Kotikov and Parente
(1999); Illarionov  _et al._ (2008); Kaidalov  _et al._ (2001); Donnachie and
Landshoff (2003). Of course, the data of structure function $F_{2}$ at very
high $Q^{2}$ are all for the large $x$ scale. All these considerations lead us
to be interested in double-asymptotic-scaling (DAS) phenomenon, which means
the limits for $Q^{2}\to\infty$ and $x\to 0$.

In present work, we associate the experiments as well as the QCD evolution
theory to analyze the proton structure function from the perspective of the BK
equation. Based on the form of analytic solution introduced in previous work
Wang  _et al._ (2021), we calculate the proton structure function $F_{2}$ at
higher $Q^{2}$ and analyze the small $x$ behavior by fitting computed results.
The organization of this paper is as follows. In Sec. II, we briefly introduce
the methods and principles which of calculating the proton structure function
$F_{2}$ by using analytical solution of BK equation in Ref. Wang  _et al._
(2021). We then discuss, in Sec. III how one can get the structure function
$F_{2}$ and corresponding effective slop parameter $\lambda$ in high energies.
In Sec. IV we briefly discuss the interesting results from Sec. III. The
summaries about this work are represented in Sec. V.

##  II Formalism

We consider the high-energy scattering process with the dipole model (see FIG.
1). The high-speed moving photon fluctuates the $q\bar{q}$ pair from QCD
vacuum into a dipole. The total cross section of the overall $\gamma^{*}p$
scattering is represented by integrating the dipole cross section with the
photon wave functions Mueller and Patel (1994); Kowalski and Teaney (2003);
Kowalski  _et al._ (2006). The momentum space formulation of the dipole
scattering cross section can be obtained from the Fourier transform. If the
transverse profile of the proton is considered as an isotropic disk with
radius $R_{p}$, the dipole scattering cross section is considered to be
proportional to the forward scattering amplitude $\mathcal{N}(r,Y)$ related to
de Santana Amaral  _et al._ (2007)

$\sigma_{\mathrm{dip}}^{\gamma^{*}p}(r,Y)=2\pi R_{p}^{2}\mathcal{N}(r,Y),$ (1)

in which $R_{p}$ is treated as the electronic radius of proton.

Figure 1: (color online). The elastic scattering of a virtual photon on a
proton in the dipole representation Kowalski  _et al._ (2006).

Based on the above discussions and some algebraic calculations, the structure
function of proton can be expressed in momentum space as Barone and Predazzi
(2002); de Santana Amaral  _et al._ (2007)

$\small
F_{2}\left(x,Q^{2}\right)=\frac{Q^{2}R_{p}^{2}N_{c}}{4\pi^{2}}\int_{0}^{\infty}\frac{dk}{k}\int_{0}^{1}dz\left|\tilde{\Psi}\left(k^{2},z;Q^{2}\right)\right|^{2}\mathcal{N}(k,Y),$
(2)

with the Fourier transform

$\small\mathcal{N}(k,Y)=\frac{1}{2\pi}\int\frac{d^{2}r}{r^{2}}e^{i\mathbf{k}\cdot\mathbf{r}}\mathcal{N}(r,Y)=\int_{0}^{\infty}\frac{dr}{r}J_{0}(kr)\mathcal{N}(r,Y).$
(3)

The photo wave functions $\tilde{\Psi}$ in Eq. (2) can be found in Ref. de
Santana Amaral  _et al._ (2007) in the momentum space. Of course, not only DIS
calculations, but also the vector meson diffractive production process has a
similar wave function representation to calculate Kowalski  _et al._ (2006);
Xie and Chen (2018). Now we shall handle the forward scattering amplitude by
the method from Wang  _et al._ (2021). The BK equation is rewritten as the
Fisher-KPP equation FISHER (1937) with some variable substitution. We use the
analytical solution of BK equation calculated by Homogeneous Balance method in
Refs. Wang (1995, 1996); Zhou  _et al._ (2003, 2004); Wang and Li (2014), thus
it is expressed as Wang  _et al._ (2021)

$\mathcal{N}(L,Y)=\frac{A_{0}e^{5A_{0}Y/3}}{\left[e^{5A_{0}Y/6}+e^{\left[-\theta+\sqrt{A_{0}/6A_{2}}\left(L-A_{1}Y\right)\right]}\right]^{2}},$
(4)

where the variable $L=\ln\left(k^{2}/k_{0}^{2}\right)$ with the cutoff
$k_{0}=\Lambda_{\mathrm{QCD}}=200$ MeV. The parameter values $\theta$,
$A_{0}$, $A_{1}$, $A_{2}$ are referred to Marquet and Soyez (2005); Wang  _et
al._ (2021), which can be obtained by fitting experimental data.

In accordance with all the discussions in this section, we shall relate the
structure function $F_{2}$ of the proton to solution Eq. (4) of the BK
equation with reference to the QCD evolution behaviors. We discuss the
analysis process in detail is as following sections.

##  III Structure function $\mathbf{F_{2}(x,Q^{2})}$ in High energies

In Ref. de Santana Amaral  _et al._ (2007), the authors already discussed in
detail the process of calculating the structure function $F_{2}$, including
the selection of parameters contained in the momentum space photon wave
function. We use the same formalism from Ref. de Santana Amaral  _et al._
(2007) to get several parameters in the analytical solution of the BK equation
by fitting HERA data Andreev  _et al._ (2014); Abramowicz  _et al._ (2015).
Considering the small $x$ limit, we chose to fit the experimental data in the
range of 2.5 GeV${}^{2}<Q^{2}<$250 GeV2, because the amount of data as well as
the precision in this region are appropriate.

According to the discussions from Wang  _et al._ (2021), we fixed the strong
coupling constant
$\bar{\alpha_{\mathrm{s}}}\equiv\alpha_{\mathrm{s}}N_{c}/\pi=0.191$. It is
effective in the energy range we considered. In order to calculate $F_{2}$
with Eq. (2), the color number $N_{c}$ is determined to $3$, and the
considerations of quark mass and flavor are not changed which we used in Ref.
de Santana Amaral  _et al._ (2007). In other words, we choose
$m_{u}=m_{d}=m_{q}=140$ MeV, $m_{c}=1.3$ GeV for quarks mass de Santana Amaral
_et al._ (2007), proton size is treated as the electronic radius $R_{p}=4.23$
GeV-1. Based on these fixed parameters options, we shall use the HERA data to
fit and obtain the parameters in Eq. (4). The fitting results are displayed in
FIG. 2 (with partial fitting results) and the fitting parameters of Eq. (4)
are determined as $A_{0}=0.5262\pm 0.0024$, $A_{1}=1.4383\pm 0.0133$,
$A_{2}=0.1047\pm 0.0008$ and $\theta=-0.4572\pm 0.0110$.

Figure 2: (color online). $x$ dependence of structure function
$F_{2}(x,Q^{2})$ with bins of $Q^{2}$. The data from H1 Andreev  _et al._
(2014) (blue squares) are compared with our fit by analytical solution of BK
Wang  _et al._ (2021) (red lines). The parameters determination is discussed
in text.

Due to all the constraints of the experiment, one is often interested in the
physics at the energy limit, such as $x\to 0$ and $Q^{2}\to\infty$. This
research we discussed is always more mathematical, but it is usually possible
to give some interesting predictions we want. We wish to investigate the
small-$x$ behavior of $F_{2}$ at higher $Q^{2}$. We can find the slope
parameter $\lambda\propto\partial\ln F_{2}/\partial\ln(1/x)$ at different
$Q^{2}$ in Ref. Cvetic  _et al._ (2009). At small-$x$ range ($x<0.01$), the
structure function of proton $F_{2}$ can be parameterized as a power like form

$F_{2}\left(x,Q^{2}\right)=Cx^{-\lambda\left(Q^{2}\right)}.$ (5)

According to our calculations, the slope parameter $\lambda$ is determined
from computed $F_{2}(x,Q^{2})$ by using BK equation and dipole amplitude, with
Eqs. (2) and (4). This will be discussed in the next section.

##  IV Results and discussions

The BK equation, as a QCD theory, has some predictive power at limit
conditions. Previous works have made theoretical predictions based on
experimental data with proper assumptions, which are very effective and
inspiring. We take this a step further by extending both the Bjorken scale $x$
and $Q^{2}$ to the limit, taking advantage of the analytical solution of the
BK equation. As we discussed, we want to see what characterizes the behavior
of $F_{2}$ under very high-$Q^{2}$. We shall easily extend $F_{2}$ to
$Q^{2}\sim\mathcal{O}(2000)$ GeV2 by just doing some numerical calculations
($2.0\times 10^{-5}<x<1.0\times 10^{-2}$). Based on the analytical solution
(4) with the fitted parameters, the behavior of slope factor $\lambda$ are
shown in FIG. 3.

Figure 3: (color online). Slope parameter $\lambda$ v.s. energy scale $Q^{2}$
. The red squares with error bar represent our calculations, determined by
Eqs. (2), (4) and (5). The Blue circles denote the extracted $\lambda$ from
Abramowicz  _et al._ (2015); Luszczak and Kowalski (2020) ($x<0.01$,
$Q^{2}<250$ GeV2) and magenta triangles are extracted from same data from HREA
Abramowicz  _et al._ (2015) ($x<0.1$, $Q^{2}<2000$ GeV2). Red line represents
the fitting result comes from Eq. (6) which describe the red points. The blue
and magenta lines are fitting results of the corresponding color points
respectively.

Figure. 3 presents the $\lambda$ values extracted from the HERA data (blue and
magenta points) Andreev  _et al._ (2014); Abramowicz  _et al._ (2015) and our
calculated results using BK equation (red solid squares), we also give
parameterized formulas to describe the $\lambda-Q^{2}$ relation, which are
written as

$\lambda(Q^{2})=a(1-\frac{b}{\ln(Q^{2}/\Lambda_{\mathrm{QCD}}^{2})})+c,$ (6)

for red and magenta points in FIG. 3 and

$\lambda(Q^{2})=A\ln\left(\frac{Q^{2}}{\Lambda_{\mathrm{QCD}}^{2}}\right)+B,$
(7)

for blue points Luszczak and Kowalski (2020), in which the QCD cutoff
$\Lambda_{\mathrm{QCD}}=200$ MeV is fixed. We use Eq. (6) to fit our
calculations and HERA data Abramowicz  _et al._ (2015) (magenta triangles,
$Q^{2}<2000$ GeV2). All parameter information is listed in TABLE. I. TABLE. II
and III, respectively.

Let us consider this formula, in contrast to the logarithmic growth of
$\lambda$ with $Q^{2}$ considered in other works Kaidalov  _et al._ (2001);
Donnachie and Landshoff (2003), we find that the $\lambda$ growth rate
decreases slowly with increasing $Q^{2}$ because of
$\mathrm{d}\lambda/\mathrm{d}Q^{2}|_{Q^{2}\to\infty}=0$. It looks like the
behavior of the running strong coupling constant in the extreme of high energy
(see Ref. Schrempp (2005) and references cited therein). From Eq. (6), we
consider a case where $Q^{2}$ approaches infinity. At this time $\lambda$
barely grows with $Q^{2}$ and the value is taken at
$\lambda(Q^{2}\to\infty)=a+c\simeq 0.41\pm 0.01$. It is probably the limit of
the value that $\lambda$ will reach. The “frozen” $\lambda$ means no change
even if $Q^{2}$ continues to rise, and the rates of variation of $x$
dependence of $F_{2}$ are fixed, too. We also consider the selection of the
fitting range of $F_{2}$ in the limit $Q^{2}$ and found no differences ($C$
and $\lambda$ in Eq. (5)) between $x<0.01$ and $x<0.001$ we selected under
this condition.

In order to explain the “frozen” $\lambda$, we consider the behavior of gluons
in high-energy $\gamma^{*}p$ scattering by using the BK equation. The
parameter $\lambda$ as the power index part represents the gluon emissions per
unit of rapidity from high speed dipole Bartels and Kowalski (2001); Luszczak
and Kowalski (2020), because of the formation of structure function is written
as Ellis  _et al._ (2008); Kowalski  _et al._ (2010)

$F_{2}\left(x,Q^{2}\right)=\int_{x}^{1}dz\int\frac{dk}{k}\Phi_{\text{DIS
}}(z,Q,k)xg\left(\frac{x}{z},k\right),$ (8)

where $xg\left(\frac{x}{z},k\right)$ denotes the unintegrated gluon density
defined from Ellis  _et al._ (2008), expressing as a power law form with index
$\lambda$ (or eigenvalue of BFKL $\omega_{n}$). On one hand, the invariant
$\lambda$ in high $Q^{2}$ region also indicates that the gluon emission rate
achieves at equilibrium. The $\lambda$’s behavior we obtained is also
generally consistent with the discussions and results in previous works Martin
_et al._ (1996); Golec-Biernat and Wusthoff (1998, 1999). On the other hand,
with increasing $Q^{2}$ and decreasing $x$ Luszczak and Kowalski (2020), the
phase space for gluon emission grows quickly with increasing, the gluons
overlap with each other when the numbers of gluon is large, the phase space
growth is limited and eventually achieves a particular value, which is also
unitarity of the QCD theory requires.

Our results are developed on the basis of previous related works Ellis  _et
al._ (2008); Kowalski  _et al._ (2010, 2011, 2013, 2014, 2016, 2017); Ellis
_et al._ (2008); Kowalski  _et al._ (2010, 2011, 2013, 2014, 2016, 2017);
Salam (1998); Cvetic  _et al._ (2009); Cooper-Sarkar  _et al._ (1998); Kotikov
and Parente (1999); Kotikov (2007); Ball and Forte (1994); Illarionov  _et
al._ (2008); Mankiewicz  _et al._ (1997), which generalize the
$Q^{2}$-dependence lambda to the higher $Q^{2}$ situation. However, in this
work, we have taken some approximations, such as the selection of fixed strong
coupling constants. This approximation means that we shall think the
calculation results more significant at some particular $Q^{2}$, while at very
low or high $Q^{2}$ there will be differences (see FIG. 3). We argue that it
is feasible to use the BK equation to investigate the proton structure
function, although we make some approximations. Our results start from the QCD
evolution theory, the BK equation has been shown to be a opening window in DIS
and diffraction processes studies Mueller (2001); Stasto  _et al._ (2001);
Munier and Peschanski (2003). The problems we encountered are explained and
solved in the future researches, which will be addressed by the solution of
the BK equation in case of running coupling constants Marquet  _et al._ (2005)
or the numerical solution of JIMWLK equation Mäntysaari and Schenke (2018),
especially DGLAP equation Ball and Forte (1994) in future.

##  V Summary

In this paper, on one hand, we mainly use the analytical solution of the BK
equation from Ref. Wang  _et al._ (2021) to calculate the proton structure
function $F_{2}$ at the higher $Q^{2}$ and discuss the behaviors of structure
function in the small-$x$ region. We find that the speed of growth of
$\lambda$, the slope parameter of $F_{2}$, decreases as $Q^{2}$ increases. It
eventually maybe converges to a certain value, precisely around $\lambda\simeq
0.41\pm 0.01$, of course, it requires large $Q^{2}$ scale. On the other hand,
we present some phenomenological discussions of the results. In our views, we
think gluons overlap and phase space growth supression at high scale could
explain the behavior of parameter $\lambda$. The conclusion will definitely
need to be tested by future high-precision Electron-Ion Collision experiments
Accardi  _et al._ (2016); Abdul Khalek  _et al._ (2021); Chen (2018); Chen
_et al._ (2020); Anderle  _et al._ (2021) in wide ranges of $Q^{2}$ and $x$.

###### Acknowledgements.

The authors are very grateful to Dr. Rong Wang for providing suggestions about
the error analysis and fruitful discussions. This work is supported by the
Strategic Priority Research Program of Chinese Academy of Sciences under the
Grant NO. XDB34030301.

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Table 1: Parameters determination according to Eq. (6) and (7), using our calculations and HERA data. The results of the last two lines come from Refs. Abramowicz  _et al._ (2015); Luszczak and Kowalski (2020). Parameters |  $\Lambda_{\mathrm{QCD}}$ (MeV) | $a$ | $b$ | $c$ | $A$ | $B$
---|---|---|---|---|---|---
This work | 200 | $0.162\pm 0.006$ | $4.800\pm 0.168$ | $0.244\pm 0.006$ |  |
Fits from Luszczak and Kowalski (2020) | 200 |  |  |  | $0.044\pm 0.001$ | $-0.020\pm 0.004$
Fits from Abramowicz  _et al._ (2015) | 200 | $0.195\pm 0.017$ | $10.657\pm 0.838$ | $0.395\pm 0.016$ |  |
Table 2: The results for the $\lambda$ constant obtained from the fits of the function (5) from Ref. Luszczak and Kowalski (2020) ($x<0.01$), $Q^{2}<250$ GeV2. $Q^{2}$ (GeV2) |  $\lambda$ |  $\delta\lambda$
---|---|---
0.35 |  0.110 |  0.008
0.4 |  0.082 |  0.009
0.5 |  0.100 |  0.009
0.65 |  0.121 |  0.011
0.85 |  0.150 |  0.014
1.2 |  0.133 |  0.013
1.5 |  0.142 |  0.009
2 |  0.159 |  0.007
2.7 |  0.169 |  0.005
3.5 |  0.173 |  0.004
4.5 |  0.189 |  0.004
6.5 |  0.200 |  0.003
8.5 |  0.208 |  0.004
10 |  0.226 |  0.007
12 |  0.215 |  0.005
15 |  0.237 |  0.003
18 |  0.242 |  0.003
22 |  0.258 |  0.007
27 |  0.267 |  0.004
35 |  0.280 |  0.003
45 |  0.292 |  0.004
60 |  0.313 |  0.005
70 |  0.332 |  0.009
90 |  0.321 |  0.007
120 |  0.352 |  0.008
150 |  0.339 |  0.011
200 |  0.373 |  0.010
250 |  0.417 |  0.014
Table 3: The results for the $\lambda$ constant obtained from the fits of the function (5) from Ref. Abramowicz  _et al._ (2015) ($x<0.1$, $Q^{2}<2000$ GeV2). $Q^{2}$ (GeV2) |  $\lambda$ |  $\delta\lambda$
---|---|---
10 |  0.224 |  0.008
12 |  0.223 |  0.006
15 |  0.240 |  0.004
18 |  0.244 |  0.005
22 |  0.269 |  0.012
27 |  0.271 |  0.005
35 |  0.284 |  0.006
45 |  0.296 |  0.007
60 |  0.316 |  0.008
70 |  0.330 |  0.014
90 |  0.299 |  0.013
120 |  0.319 |  0.019
150 |  0.334 |  0.0281
200 |  0.339 |  0.006
250 |  0.347 |  0.007
300 |  0.361 |  0.008
400 |  0.373 |  0.010
500 |  0.396 |  0.023
650 |  0.381 |  0.014
800 |  0.397 |  0.025
1000 |  0.380 |  0.039
1200 |  0.425 |  0.028
1500 |  0.364 |  0.047
2000 |  0.345 |  0.105


# Invariant measures for reducible generalized Bratteli diagrams

Sergey Bezuglyi  Department of Mathematics, University of Iowa, Iowa City, IA
52242-1419 USA,
ORCID: 0000-0002-3212-1750<EMAIL_ADDRESS>,  Olena Karpel  AGH
University of Krakow, Faculty of Applied Mathematics, al. Adama Mickiewicza
30, 30-059 Kraków, Poland & B. Verkin Institute for Low Temperature Physics
and Engineering, 47 Nauky Ave., Kharkiv, 61103, Ukraine
ORCID: 0000-0001-5988-3774<EMAIL_ADDRESS>and  Jan Kwiatkowski  Faculty
of Mathematics and Computer Science, Nicolaus Copernicus University, ul.
Chopina 12/18, 87-100 Toruń, Poland<EMAIL_ADDRESS>
###### Abstract.

In 2010, Bezuglyi, Kwiatkowski, Medynets and Solomyak [Ergodic Theory Dynam.
Systems 30 (2010), no.4, 973-1007] found a complete description of the set of
probability ergodic tail invariant measures on the path space of a standard
(classical) stationary reducible Bratteli diagram. It was shown that every
distinguished eigenvalue for the incidence matrix determines a probability
ergodic invariant measure. In this paper, we show that this result does not
hold for stationary reducible generalized Bratteli diagrams. We consider
classes of stationary and non-stationary reducible generalized Bratteli
diagrams with infinitely many simple standard subdiagrams, in particular, with
infinitely many odometers as subdiagrams. We characterize the sets of all
probability ergodic invariant measures for such diagrams and study partial
orders under which the diagrams can support a Vershik homeomorphism.

###### Key words and phrases:

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

###### 2020 Mathematics Subject Classification:

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

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Basic definitions on Bratteli diagrams
    2. 2.2 Subdiagrams and measure extension
    3. 2.3 Invariant measures on stationary standard Bratteli diagrams
    4. 2.4 Invariant measures on stationary irreducible generalized Bratteli diagrams
  3. 3 Measure extension for generalized Bratteli diagrams
  4. 4 Reducible Bratteli diagrams with infinitely many odometers
  5. 5 Some classes of generalized Bratteli diagrams with infinitely many odometers
    1. 5.1 Stationary generalized Bratteli diagrams with infinitely many odometers.
    2. 5.2 Non-stationary generalized Bratteli diagrams with infinitely many odometers.
  6. 6 Vershik maps on generalized Bratteli diagrams with infinitely many odometers

##  1\. Introduction

Bratteli diagrams provide an important tool for the study of dynamical systems
in measurable, Cantor, and Borel dynamics. A generalized Bratteli diagram is a
natural extension of the notion of a classical (standard) Bratteli diagram,
where each level has a countably infinite set of vertices. While standard
Bratteli diagrams are particularly useful in Cantor dynamics to describe the
simplex of probability tail invariant measures and study other properties of
dynamical systems (see e.g. surveys [Dur10, BK16, BK20, Put18]), generalized
Bratteli diagrams are used to model non-compact Borel dynamical systems
[BDK06]. Recent papers developed the study of dynamical systems on generalized
Bratteli diagrams [BJ22, BJS22, BJKS23]. It was shown that generalized
Bratteli diagrams have many interesting phenomena in comparison with the
standard case. For example, one can find stationary generalized Bratteli
diagrams with either a unique probability invariant measure, or uncountable
many probability invariant measures, or without such measures at all.

This paper continues the study of generalized Bratteli diagrams and focuses
mostly on a class of stationary reducible generalized Bratteli diagrams. We
consider tail invariant measures and the existence of a Vershik map on such
diagrams. The case of irreducible generalized Bratteli diagrams was considered
in [BJ22, BJKS23].

The importance of the case of stationary Bratteli diagrams is based on the
following facts. Stationary standard Bratteli diagrams provide models of
substitution dynamical systems (minimal and non-minimal ones), see [DHS99],
[BKM09]. The case of stationary generalized Bratteli diagrams is more
challenging to study. Right now, it is known that a class of substitution
dynamical systems on an infinite alphabet (considered first by [Fer06]) can be
realized as a generalized Bratteli diagram [BJS22].

Having such a duality between the diagrams and substitution dynamical systems,
one can answer the principal question about an explicit description of the set
of ergodic probability measures. In [BKMS10], the authors found a transparent
algorithm for the construction of ergodic probability measures. For this, one
needs to find all distinguished Perron eigenvalues of the incidence matrix.
Then using the corresponding eigenvector, the values of the invariant measure
on the cylinder sets are determined by a simple formula (see Theorem 2.11 for
details).

It seems rather surprising, but the mentioned theorem does not hold for
stationary generalized Bratteli diagrams even in the case when all simple
subdiagrams are standard odometers. Instead, we state a new result that
provides a required characterization of probability tail invariant measures.
For this, we use the method developed in [BKK15, ABKK17], where ergodic
invariant measures on standard Bratteli diagrams were obtained by using a
procedure of an extension from a subdiagram. In this paper, we show that the
same method can be used for generalized Bratteli diagrams. Moreover, for the
class of the so-called reducible Bratteli diagrams with infinitely many
odometers, this method gives all probability ergodic invariant measures on the
diagram.

The outline of the paper is as follows. Section 2 provides main facts
concerning standard and generalized Bratteli diagrams, the procedure of
measure extension, and the description of all probability ergodic invariant
measures for reducible stationary standard Bratteli diagrams. In Section 3, we
briefly discuss the procedure of measure extension from a vertex subdiagram
for generalized Bratteli diagrams. In Section 4, we introduce the notion of
reducible generalized Bratteli diagrams with infinitely many odometers and
focus on the study of tail invariant measures and their extensions. We give a
complete classification of probability ergodic invariant measures on such
diagrams. In Section 5, we present various classes of stationary and non-
stationary reducible generalized Bratteli diagrams with infinitely many
odometers and characterize their sets of probability ergodic invariant
measures. Section 6 is devoted to different orders on reducible generalized
Bratteli diagrams with infinitely many odometers. For different classes of
orders on such diagrams, we study whether the corresponding Vershik map can be
extended to a homeomorphism. Our main results are presented in Theorems 3.1,
4.3, 5.5, 5.8, 6.2.

##  2\. Preliminaries

In this section, we recall the main definitions and facts concerning invariant
measures on standard and generalized stationary Bratteli diagrams. For more
details see [BKMS10], [BK16], [BJ22], [BJKS23].

Throughout the paper, 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 $|\cdot|$ for the cardinality of a set.

###  2.1. Basic definitions on Bratteli diagrams

######  Definition 2.1.

A (standard) Bratteli diagram is an infinite graph $B=(V,E)$ such that the
vertex set $V=\bigsqcup_{i\geq 0}V_{i}$ and the edge set $E=\bigsqcup_{i\geq
0}E_{i}$ are partitioned into disjoint subsets $V_{i}$ and $E_{i}$, where

(i) $V_{0}=\\{v_{0}\\}$ is a single point;

(ii) $V_{i}$ and $E_{i}$ are finite sets for all $i$;

(iii) there exists a range map $r\colon E\rightarrow V$ and a source map
$s\colon E\rightarrow V$ such that $r(E_{i})=V_{i+1}$ and $s(E_{i})=V_{i}$ for
all $i\geq 1$.

######  Definition 2.2.

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

$(i)$ For every $i\in\mathbb{N}_{0}$, the number of vertices at each level
$V_{i}$ is countably infinite, and 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
$i\in\mathbb{N}_{0}$.

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

Let $B=(V,E)$ be a standard or generalized Bratteli diagram. We will call the
set $V_{i}$ the $i$th level of the diagram $B$. For generalized Bratteli
diagrams, we will identify each $V_{i}$ with $\mathbb{N}$. 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.
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. The path space $X_{B}$ is a
zero-dimensional Polish space.

For vertices $v\in V_{m}$ and $w\in V_{n}$, 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
Bratteli diagram $B=(V,E)$ a sequence of non-negative 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 the case of a generalized Bratteli diagram, incidence matrices are
countably infinite.

For $w\in V_{n}$ and $n\in\mathbb{N}_{0}$, denote

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

The collection $(X_{w}^{(n)})_{w}$ of all such sets forms a partition
$\zeta_{n}$ of $X_{B}$ into Kakutani-Rokhlin towers corresponding to the
vertices from $V_{n}$. For $w\in V_{n}$ and $v_{0}\in V_{0}$, we set

$h^{(n)}_{v_{0},w}=|E(v_{0},w)|$

and define

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

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

######  Definition 2.3.

Given a standard or generalized Bratteli diagram $B=(V,E)$ and a monotone
increasing sequence $(n_{k}:k\in\mathbb{N}_{0})$ with $n_{0}=0$, we define a
new 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 $B=(V,E)$.

######  Remark 2.4.

Notice that each telescoping of a generalized Bratteli diagram is again a
generalized Bratteli diagram.

######  Definition 2.5.

(1) Let $B=B(F_{n})$, where $(F_{n})_{n}$ are the incidence matrices of $B$,
be a standard or generalized Bratteli diagram. If $F_{n}=F$ for every
$n\in\mathbb{N}$ (for standard diagrams) or every $n\in\mathbb{N}_{0}$ (for
generalized diagrams), then the diagram $B$ is called stationary. We will
write $B=B(F)$ in this case. Unless stated otherwise, we will assume that
every standard Bratteli diagram has a ‘‘simple hat’’, which means that there
is a single edge from the vertex $v_{0}$ to every vertex $v\in V_{1}$.

(2) A standard Bratteli diagram is called simple if there exists a telescoping
$B^{\prime}$ of $B$ such that all entries of the incidence matrices of
$B^{\prime}$ are positive. Since every vertex of a generalized Bratteli
diagram can have only finitely many incoming edges, the notion of a simple
Bratteli diagram cannot be applied to generalized diagrams.

(3) A generalized Bratteli diagram $B=(V,E)$ 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.

We will consider tail-invariant measures on the path space $X_{B}$ of a
standard or generalized Bratteli diagram $B$. By term measure we always mean a
non-atomic positive Borel measure.

######  Definition 2.6.

Let $B$ be a standard or generalized Bratteli diagram. Two paths $x=(x_{i})$
and $y=(y_{i})$ in $X_{B}$ are called tail equivalent if there exists
$n\in\mathbb{N}_{0}$ 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
$X_{B}$ (every equivalence class is countable and $\mathcal{R}$ is a Borel
subset of $X_{B}\times X_{B}$) which is called the tail equivalence relation.
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
$\mu([\overline{e}])=\mu([\overline{e}^{\prime}])$.

######  Theorem 2.7.

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

  1. (1)

Let $\mu$ be a tail-invariant measure on $B$ which takes finite values on
cylinder sets. For every $n\in\mathbb{N}_{0}$, define the vector
$\overline{p}^{(n)}=\langle p^{(n)}_{w}:w\in V_{n}\rangle$, where

(2.2) $p^{(n)}_{w}=\mu([\overline{e}(w)]),\ \ w\in V_{n}.$

where $[\overline{e}(w)]$ is a cylinder set which ends in the vertex $w$. Then
the vectors $\overline{p}^{(n)}$ satisfy the relation

(2.3) $F_{n}^{T}\overline{p}^{(n+1)}=\overline{p}^{(n)},\quad n\geq 0.$

  2. (2)

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

The proof of Theorem 2.7 is straightforward and can be found in [BKMS10] (for
classical Bratteli diagrams) and [BJ22] (for generalized Bratteli diagrams).

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 $E_{i},\ i=0,1,...$, where edges $e,e^{\prime}$ are comparable if and
only if $r(e)=r(e^{\prime})$.

######  Definition 2.8.

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 a (finite or infinite) path $e=(e_{i})$ maximal (respectively minimal)
if every $e_{i}$ is maximal (respectively minimal) among all elements from
$r^{-1}(r(e_{i}))$. Denote by $X_{max}$ ($X_{min}$) the sets of all infinite
maximal (minimal) paths in $X_{B}$.

######  Definition 2.9.

For an ordered generalized Bratteli diagram $B=(V,E,>)$, we define a Borel
transformation $\varphi_{B}:X_{B}\setminus X_{max}\rightarrow X_{B}\setminus
X_{min}$ 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 Borel 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 2.10.

If the cardinalities of $X_{max}$ and $X_{min}$ are the same then there always
exists a Borel extension of $\varphi_{B}$ to the whole path space $X_{B}$.

In general, every measure $\mu$ that is invariant with respect to a Vershik
map is also tail-invariant. If the sets of $X_{min}$ and $X_{max}$ have zero
measure, then we can identify tail-invariant measures with measures invariant
with respect to a Vershik map.

###  2.2. Subdiagrams and measure extension

In this subsection, we give the basic definitions and results on subdiagrams
of standard and generalized Bratteli diagrams. We also describe the notion of
measure extension. We use the approach developed first in [BKK15] and
[ABKK17].

Let $B=(V,E)$ be a standard or generalized Bratteli diagram. Consider nonempty
subsets $\overline{V}\subset V$ and $\overline{E}\subset E$ that can be
written as $\overline{V}=\bigcup_{n}\overline{V}_{n}$ and
$\overline{E}=\bigcup_{n}\overline{E}_{n}$, where $\overline{V}_{n}\subset
V_{n}$ and $\overline{E}_{n}\subset E_{n}$. Then we say that the pair
$\overline{B}=(\overline{V},\overline{E})$ defines a subdiagram of $B$ if
$\overline{V}=s(\overline{E})$ and
$s(\overline{E})=r(\overline{E})\cup\overline{V}_{0}$.

To define a vertex subdiagram of $B$, we begin with a sequence
$\overline{W}=\\{W_{n}\\}_{n>0}$ of proper nonempty subsets $W_{n}$ of
$V_{n}$, and set $W^{\prime}_{n}=V_{n}\setminus W_{n}\neq\emptyset$ for all
$n$. The vertex subdiagram $\overline{B}=(\overline{W},\overline{E})$ is
formed by the vertices from $W_{n}$ and by the set of edges $\overline{E}_{n}$
whose source and range are in $W_{n}$ and $W_{n+1}$, respectively. Thus, the
incidence matrix $\overline{F}_{n}$ of $\overline{B}$ has the size
$|W_{n+1}|\times|W_{n}|$, and it is represented by a block of $F_{n}$
corresponding to the vertices from $W_{n}$ and $W_{n+1}$.

The path space $X_{\overline{B}}$ of a vertex subdiagram $\overline{B}$ is a
closed subset of $X_{B}$. On the other hand, there are closed subsets of
$X_{B}$ which are not obtained as the path space of a Bratteli subdiagram. It
was proved in [GPS04] that a closed subset $Z\subset X_{B}$ is the path space
of a subdiagram if and only if $\mathcal{R}|_{Z\times Z}$ is an etalé
equivalence relation (see [GPS04] for details).

Below we explain the procedure of measure extension from a subdiagram. This
procedure was considered earlier in [BKK15] and [ABKK17] for standard Bratteli
diagrams, but it works also for generalized diagrams. The diagram $B$ below
can be either a standard or generalized Bratteli diagram. Let
$\widehat{X}_{\overline{B}}:=\mathcal{R}(X_{\overline{B}})$ be the subset of
all paths in $X_{B}$ that are tail equivalent to paths from
$X_{\overline{B}}$. In other words, $\widehat{X}_{\overline{B}}$ is the
smallest $\mathcal{R}$-invariant subset of $X_{B}$ containing
$X_{\overline{B}}$. Let $\overline{\mu}$ be an ergodic tail invariant
probability measure on $X_{\overline{B}}$. Then $\overline{\mu}$ can be
canonically extended to the ergodic measure $\widehat{\overline{\mu}}$ on the
space $\widehat{X}_{\overline{B}}$ by tail invariance, see [BKMS13], [ABKK17].
More specifically, let the measure $\overline{\mu}$ be defined by a sequence
of positive vectors $\\{\overline{p}^{(n)}:n\in\mathbb{N}_{0}\\}$ satisfying
Theorem 2.7, that is
$\overline{F}_{n}(\overline{p}^{(n+1)})=\overline{p}^{(n)},n\in\mathbb{N}_{0}$,
where $\overline{F}_{n}$ is the incidence matrix for the subdiagram
$\overline{B}$. Then, for every cylinder set $[\overline{e}]\subset X_{B}$
with $r(\overline{e})=v\in\overline{V}_{n}$, we set
$\widehat{\overline{\mu}}([\overline{e}])=\overline{p}^{(n)}_{v}$. Then
$\widehat{\overline{\mu}}$ is defined on all clopen sets and it can be finally
extended to a Borel measure on of $X_{B}$.

Let $\overline{B}$ be a vertex subdiagram of a generalized Bratteli diagram
$B$ defined by a sequence of subsets $(W_{i})$. Denote by
$\widehat{X}_{\overline{B}}^{(n)}$ the set of all paths
$x=(x_{i})_{i=1}^{\infty}$ from $X_{B}$ such that the finite path
$(x_{1},\ldots,x_{n})$ ends at a vertex $v$ of $\overline{B}$, and the tail
$(x_{n+1},x_{n+2},\ldots)$ belongs to $\overline{B}$, i.e.,

(2.4)
$\widehat{X}_{\overline{B}}^{(n)}=\\{x=(x_{i})\in\widehat{X}_{\overline{B}}:r(x_{i})\in
W_{i},\ \forall i\geq n\\}.$

It is obvious that
$\widehat{X}_{\overline{B}}^{(n)}\subset\widehat{X}_{\overline{B}}^{(n+1)}$,
$\widehat{X}_{\overline{B}}=\bigcup_{n}\widehat{X}_{\overline{B}}^{(n)}$, and

(2.5)
$\widehat{\overline{\mu}}(\widehat{X}_{\overline{B}})=\lim_{n\to\infty}\widehat{\overline{\mu}}(\widehat{X}_{\overline{B}}^{(n)})=\lim_{n\to\infty}\sum_{w\in
W_{n}}H^{(n)}_{w}\overline{p}^{(n)}_{w}.$

This limit can be finite or infinite. If it is finite, then we say that
$\overline{\mu}$ admits a finite measure extension
$\widehat{\overline{\mu}}(\widehat{X}_{\overline{B}})<\infty$. To obtain an
ergodic invariant measure on the whole path space $X_{B}$, we set
$\widehat{\overline{\mu}}(X_{B}\setminus\widehat{X}_{\overline{B}})=0$.

###  2.3. Invariant measures on stationary standard Bratteli diagrams

In this subsection, we recall the explicit description of all probability
ergodic invariant measures on stationary standard Bratteli diagrams, the
exposition is based on [BKMS10].

Let $B$ be a stationary standard Bratteli diagram with $N$ vertices on each
level $n\geq 1$. We identify every set of vertices $V_{n}$ for $n\geq 1$ with
the set $\\{1,\ldots,N\\}$. Denote by $F$ the corresponding incidence matrix
and let $A=F^{T}$.

One can associate to $A$ a directed graph $G(A)$ with the vertices
$\\{1,\ldots,N\\}$ such that there is an arrow from $i$ to $j$ if and only if
$a_{ij}>0$. We will say that vertices $i$ and $j$ are equivalent if either
$i=j$ or there are paths in $G(A)$ from $i$ to $j$ and from $j$ to $i$. Denote
by $\mathcal{E}_{i}$, $i=1,\ldots,m$ the corresponding equivalence classes.
Then each class $\mathcal{E}_{i}$ defines a submatrix $A_{i}$ of $A$ obtained
by restricting $A$ to the set of vertices from $\mathcal{E}_{i}$. Identify the
family of sets $\\{\mathcal{E}_{i}\\}_{i=1}^{m}$ with the set
$\\{1,\ldots,m\\}$, and define the partial order on $\\{1,\ldots,m\\}$ as
follows: for $\alpha,\beta\in\\{1,\ldots,m\\}$, we have $\beta\succeq\alpha$
if either $\alpha=\beta$ or there is a path in $G(A)$ from a vertex in
$\mathcal{E}_{\beta}$ to a vertex in $\mathcal{E}_{\alpha}$. In this case, we
say that class $\beta$ has access to class $\alpha$. If $\beta\succeq\alpha$
and $\beta\neq\alpha$, we write $\beta\succ\alpha$. The partial order $\succ$
defines the reduced directed graph $R(A)$ of $G(A)$ as follows: the set of
vertices of $R(A)$ is $\\{1,\ldots,m\\}$, there is an edge from a vertex
$\beta$ to a vertex $\alpha$ if and only if $\beta\succ\alpha$. One can
enumerate the vertices of $B$ and classes in $\\{1,\ldots,m\\}$ in such a way
that $A$ assumes a block triangular form and $\beta\succ\alpha$ if
$\alpha>\beta$ (with the usual ordering on integers):

$A=\begin{pmatrix}A_{1}&0&\ldots&0&Y_{1,s+1}&\ldots&Y_{1,m}\\\
0&A_{2}&\ldots&0&Y_{2,s+1}&\ldots&Y_{2,m}\\\
\vdots&\vdots&\ddots&\vdots&\vdots&\ddots&\vdots\\\
0&0&\ldots&A_{s}&Y_{s,s+1}&\ldots&Y_{s,m}\\\
0&0&\ldots&0&A_{s+1}&\ldots&Y_{s+1,m}\\\
\vdots&\vdots&\ddots&\vdots&\vdots&\ddots&\vdots\\\
0&0&\ldots&0&0&\ldots&A_{m}\\\ \end{pmatrix},$

where $s\geq 1$, the square matrices $\\{A_{i}\\}_{i=1}^{s}$ on the main
diagonal are non-zero irreducible matrices, each square matrix
$\\{A_{i}\\}_{i=s+1}^{m}$ is either irreducible or a $1\times 1$ zero matrix.
For any $j=s+1,\ldots,m$, at least one of the matrices
$\\{Y_{k,j}\\}_{k=1}^{j-1}$ is non-zero. Moreover, the matrix $Y_{k,j}$ is
non-zero if and only if there is an edge in $R(A)$ from $k$ to $j$. Further,
we telescope $B$ so that each non-zero matrix $A_{i}$ on the main diagonal is
strictly positive. For more details see for instance §4.4 in [LM95], where
$R(F)$ is called the graph of communicating classes.

Let $\rho_{\alpha}$ be a spectral radius (a Perron eigenvalue) of
$A_{\alpha}$. A vertex (class) $\alpha\in\\{1,\ldots,m\\}$ is called
distinguished if $\rho_{\alpha}>\rho_{\beta}$ whenever $\beta\succ\alpha$. In
particular, vertices $\\{1,\ldots,s\\}$ are distinguished vertices in $R(A)$
and $A_{\alpha}\neq 0$ for a distinguished class $\alpha$. Let
$\overline{B}_{\alpha}$ be a simple stationary subdiagram of $B$ generated by
vertices that belong to $\mathcal{E}_{\alpha}$. We call a real number
$\lambda$ a distinguished eigenvalue for $A$ if there exists a non-negative
eigenvector x with $A\textbf{x}=\lambda\textbf{x}$. A real number $\lambda$ is
a distinguished eigenvalue if and only if there is a distinguished class
$\alpha$ in $R(A)$ such that $\rho_{\alpha}=\lambda$. The corresponding non-
negative eigenvector $(\xi_{\alpha}(1),\ldots,\xi_{\alpha}(N))^{T}$ is unique
(up to scaling) and $\xi_{\alpha}(i)>0$ if and only if $i$ has access to
$\alpha$ (see [Vic85, Sch86, TS01]). The vector
$(\xi_{\alpha}(1),\ldots,\xi_{\alpha}(N))^{T}$ it is called a distinguished
eigenvector corresponding to a distinguished eigenvalue $\lambda_{\alpha}$.

We sum up the results from [BKMS10] in the following theorem:

######  Theorem 2.11.

[BKMS10] Let $B$ be a stationary standard Bratteli diagram and let $A$ be an
$N\times N$ matrix which is the transpose of the incidence matrix. Then every
probability ergodic invariant measure on $X_{B}$ corresponds to a
distinguished class of vertices in $R(A)$, a distinguished right eigenvector
for $A$, and a corresponding distinguished eigenvalue for $A$. Conversely,
every distinguished class of vertices in $R(A)$, distinguished right
eigenvector and a corresponding distinguished eigenvalue for $A$ generate a
probability ergodic invariant measure on $B$ in the following way: if
$\mu_{\alpha}$ is a probability ergodic invariant measure corresponding to a
distinguished class of vertices $\alpha$ in $R(A)$ then $\mu_{\alpha}$ is (up
to constant multiple) the extension of a unique invariant measure
$\overline{\mu}_{\alpha}$ from $\overline{B}_{\alpha}$. If
$(\xi_{\alpha}(1),\ldots,\xi_{\alpha}(N))^{T}$ and $\lambda_{\alpha}$ are the
corresponding distinguished eigenvector and eigenvalue then

$\mu_{\alpha}([\overline{e}(v_{0},w)])=\frac{\xi_{\alpha}(w)}{\lambda_{\alpha}^{n-1}},$

where $[\overline{e}(v_{0},w)]$ is a cylinder set corresponding to a finite
path $\overline{e}(v_{0},w)$ which ends in a vertex $w$ on level $n\geq 1$.

###  2.4. Invariant measures on stationary irreducible generalized Bratteli
diagrams

In [BJ22, BJKS23], the authors used the Perron-Frobenius theory for infinite
matrices to describe tail invariant measures on the path space of a class of
irreducible stationary generalized Bratteli diagrams. In particular, it was
shown that

######  Theorem 2.12 ([BJKS23]).

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 by the
formula

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

where $[\overline{e}(w,v)]$ is the cylinder set which corresponds to a finite
path $\overline{e}(w,v)$ that begins at $w\in V_{0}$ and ends at $v\in V_{n}$,
$n\in\mathbb{N}$, is the unique probability $\varphi_{B}$-invariant measure
that takes positive values on cylinder sets.

##  3\. Measure extension for generalized Bratteli diagrams

In this section, we consider the procedure of a measure extension from a
vertex subdiagram for generalized Bratteli diagrams. The proof is essentially
the same as in [ABKK17].

######  Theorem 3.1.

Let $B$ be a generalized Bratteli diagram $B=(V,E)$ with incidence matrices
$(F_{n})$. Let $\overline{B}$ be a (standard or generalized) vertex subdiagram
of $B$ determined by a sequence $(W_{n})$ of proper subsets $W_{n}\subset
V_{n}$ for each $n\in\mathbb{N}$. Let $\overline{\mu}$ be a probability
measure on the path space $X_{\overline{B}}$ of $\overline{B}$. Then

(3.1)
$\widehat{\overline{\mu}}(\widehat{X}_{\overline{B}})=1+\sum_{n=0}^{\infty}\sum_{v\in
W_{n+1}}\sum_{w\in{W}^{\prime}_{n}}{f}_{vw}^{(n)}H_{w}^{(n)}\overline{p}_{v}^{(n+1)},$

where $W_{n}^{{}^{\prime}}=V_{n}\setminus W_{n},\ n=0,1,2,\ldots$.

In particular, the following statements are equivalent:

(i)
$\displaystyle\widehat{\overline{\mu}}(\widehat{X}_{\overline{B}})<\infty,$

(ii) $\displaystyle\sum_{n=0}^{\infty}\sum_{v\in
W_{n+1}}\sum_{w\in{W}^{\prime}_{n}}{f}_{vw}^{(n)}H_{w}^{(n)}\overline{p}_{v}^{(n+1)}<\infty.$

###### Proof.

To prove the theorem, fix $n$ and begin with equality (2.5). We have

$\displaystyle\widehat{\overline{\mu}}(\widehat{X}_{\overline{B}}^{(n+1)})=$
$\displaystyle\ \sum_{v\in
W_{n+1}}\overline{p}_{v}^{(n+1)}H_{v}^{(n+1)}=\sum_{v\in
W_{n+1}}\overline{p}_{v}^{(n+1)}\sum_{w\in V_{n}}{f}_{vw}^{(n)}H_{w}^{(n)}$
$\displaystyle=$ $\displaystyle\ \sum_{v\in
W_{n+1}}\overline{p}_{v}^{(n+1)}\sum_{w\in
W_{n}}{f}_{vw}^{(n)}H_{w}^{(n)}+\sum_{v\in
W_{n+1}}\overline{p}_{v}^{(n+1)}\sum_{w\in
W^{\prime}_{n}}{f}_{vw}^{(n)}H_{w}^{(n)}$ $\displaystyle=$ $\displaystyle\
\sum_{w\in W_{n}}H_{w}^{(n)}\sum_{v\in
W_{n+1}}\overline{f}_{vw}^{(n)}\overline{p}_{v}^{(n+1)}+\sum_{v\in
W_{n+1}}\sum_{w\in
W^{\prime}_{n}}{f}_{vw}^{(n)}\overline{p}_{v}^{(n+1)}H_{w}^{(n)}$
$\displaystyle=$ $\displaystyle\ \sum_{w\in
W_{n}}H_{w}^{(n)}\overline{p}_{w}^{(n)}+\sum_{v\in W_{n+1}}\sum_{w\in
W^{\prime}_{n}}{f}_{vw}^{(n)}\overline{p}_{v}^{(n+1)}H_{w}^{(n)}$
$\displaystyle=$ $\displaystyle\
\widehat{\overline{\mu}}(\widehat{X}_{\overline{B}}^{(n)})+\sum_{v\in
W_{n+1}}\sum_{w\in
W^{\prime}_{n}}{f}_{vw}^{(n)}\overline{p}_{v}^{(n+1)}H_{w}^{(n)}.$

Since

$\widehat{\overline{\mu}}(\widehat{X}_{\overline{B}})=1+\sum_{n\geq
0}\left(\widehat{\overline{\mu}}(\widehat{X}_{\overline{B}}^{(n+1)})-\widehat{\overline{\mu}}(\widehat{X}_{\overline{B}}^{(n)})\right),$

we get the result. ∎

######  Remark 3.2.

Note that we can also compute the measure $\widehat{\overline{\mu}}$ of any
cylinder set $[\overline{e}]$ by carefully examining which cylinder subsets
that end in the vertices of $\overline{B}$ are contained in $[\overline{e}]$.
The sum of measures of these subsets will give us the measure of
$[\overline{e}]$. In Section 5, we present such computations for concrete
examples of Bratteli diagrams.

##  4\. Reducible Bratteli diagrams with infinitely many odometers

Consider the following class of non-stationary reducible generalized Bratteli
diagrams. In this section, we will focus on the study of their tail invariant
measures. Every diagram in this class contains infinitely many odometers as
subdiagrams, which are connected by single edges.

Let the Bratteli diagram $B=B_{IO}$ be defined by the sequence of incidence
matrices

(4.1) ${F}_{n}=\begin{pmatrix}a_{n}^{(1)}&1&0&0&\ldots&\ldots&\ldots&\ldots\\\
0&a_{n}^{(2)}&1&0&0&\ldots&\ldots&\ldots\\\
0&0&a_{n}^{(3)}&1&0&0&\ldots&\ldots\\\ 0&0&0&a_{n}^{(4)}&1&0&0&\ldots\\\
\ldots&\ldots&\ldots&\ldots&\ldots&\ldots&\ldots&\ldots\\\
\end{pmatrix},\qquad n\in\mathbb{N}_{0},$

where the natural numbers $a_{n}^{(i)}\geq 2$ for all $n\in\mathbb{N}_{0}$ and
$i\in\mathbb{N}$. The index $n$ points out at the $n$-th level of the diagram
$B$, and $i$ corresponds to the number of a vertex inside $V_{n}$.

The diagram $B_{IO}$ has a natural set of elementary vertex subdiagrams
$\overline{B}_{i}$ consisting of vertical odometers where $i$ runs over the
set $\mathbb{N}$. There are exactly $a_{n}^{(i)}$ edges connecting the
vertices $i\in V_{n}$ and $i\in V_{n+1}$. We call $B$ the ‘‘diagram of
infinite odometers (DIO)’’.

Analyzing the paths space of (DIO) it is not hard to prove the following
claim:

######  Claim 4.1.

The sets $\widehat{X}_{\overline{B}_{i}}$, $i=1,2,\ldots$, are pairwise
disjoint and $X_{B}=\bigsqcup_{i=1}^{\infty}\widehat{X}_{\overline{B}_{i}}$,
where $X_{B}$ is the set of all infinite paths of (DIO).

The subdiagram $\overline{B}_{i}$ of $B$ admits a unique tail invariant
probability measure $\overline{\mu}_{i}$ on the path space
$X_{\overline{B}_{i}}$ such that for a cylinder set
$[\overline{e}]=[e_{0},\ldots,e_{n}],s(e_{j})=r(e_{j})=i$, we have

$\overline{\mu}_{i}([\overline{e}])=\frac{1}{a^{(i)}_{0}\ \cdots\
a^{(i)}_{n}}.$

The measure extension procedure applied to $\overline{B}_{i}$ gives us the
measure $\widehat{\overline{\mu}}_{i}$ on the tail invariant set
$\widehat{X}_{\overline{B}_{i}}$. It follows from Theorem 3.1 that

$\widehat{\overline{\mu}}_{i}(\widehat{X}_{\overline{B}_{i}})<\infty\
\Longleftrightarrow\ \sum_{n=0}^{\infty}\frac{H^{(n)}_{i+1}}{a^{(i)}_{0}\
\cdots\ a^{(i)}_{n}}<\infty.$

Thus, it follows from the construction of $B_{IO}$ that there are infinitely
many ergodic measures $\widehat{\overline{\mu}}_{i}$ on the path space
$X_{B}$. Some of them may be finite, while others are infinite. We will give
exact examples below. Moreover, the measures $\widehat{\overline{\mu}}_{i}$
and $\widehat{\overline{\mu}}_{j}$ are mutually singular ($i\neq j)$ because
they are supported by non-intersecting tail invariant sets
$\widehat{X}_{\overline{B}_{i}}$ and $\widehat{X}_{\overline{B}_{j}}$ (see
Claim 4.1).

Our goal is to show that there are no other ergodic measures.

######  Remark 4.2.

Let $\theta$ be a finite tail invariant measure on
$\widehat{X}_{\overline{B}_{i}}$. Then there is a constant $C>0$ such that
$\theta=C\widehat{\overline{\mu}}_{i}$.

Indeed, let $C=\theta(X_{\overline{B}_{i}})$, then
$\overline{\theta}:=\theta|_{X_{\overline{B}_{i}}}=C\overline{\mu}_{i}$ by
uniqueness of $\overline{\mu}_{i}$ on $X_{\overline{B}_{i}}$. By tail
invariance, for every $n\in\mathbb{N}_{0}$ and every cylinder set
$[e_{0},\ldots,e_{n}]$ such that $r(e_{n})=i\in V_{n+1}$ we have

$\theta([e_{0},\ldots,e_{n}])=\frac{C}{a^{(i)}_{0}\ \cdots\
a^{(i)}_{n}}=C\widehat{\overline{\mu}_{i}}([e_{0},\ldots,e_{n}]).$

######  Theorem 4.3.

Let $\mathcal{M}$ be the family of measures obtained by normalization of
measures $\widehat{\overline{\mu}}_{i}$ such that
$\widehat{\overline{\mu}}_{i}(\widehat{X}_{\overline{B}_{i}})<\infty$. Then
$\mathcal{M}$ coincides with the set of all ergodic probability tail invariant
measures on the path space $X_{B}$ of the diagram $B$.

###### Proof.

We first note that $X_{B}$ can be partitioned into the union of tail invariant
sets $\widehat{X}_{\overline{B}_{i}}$. Every $\widehat{X}_{\overline{B}_{i}}$
is an $F_{\sigma}$-set, hence it is Borel. Let $\theta$ be a finite tail
invariant measure on $X_{B}$. The support of $\theta$ is the union of some
sets $\widehat{X}_{\overline{B}_{i}}$. Then
$\theta(\widehat{X}_{\overline{B}_{i}})<\infty$ for all such $i$’s. Let
$\theta_{i}$ be the measure $\theta$ restricted to the set
$\widehat{X}_{\overline{B}_{i}}$. By Remark 4.2, every $\theta_{i}$ is
proportional to $\widehat{\overline{\mu}}_{i}$, i.e.,
$\theta_{i}=C_{i}\widehat{\overline{\mu}}_{i}$. We observe that
$\sum_{i}C_{i}=\theta(X_{B})$. This means that $\theta$ is a linear
combination of ergodic measures $\widehat{\overline{\mu}}_{i}$. ∎

######  Remark 4.4.

The same approach works in the case when the vertical odometers are replaced
with simple stationary standard Bratteli diagrams $\overline{B}_{i}$. As for
odometers, we will have a unique ergodic probability measure
$\overline{\mu}_{i}$ on the path space $X_{\overline{B}_{i}}$ defined in
Theorem 2.11. Assuming that the extension
$\widehat{\overline{\mu}}_{i}(\widehat{X}_{\overline{B}_{i}})$ is finite, we
get that this measure is unique (up to a constant). The same reasoning as in
the proof of Theorem 4.3 can be repeated.

##  5\. Some classes of generalized Bratteli diagrams with infinitely many
odometers

In this section, we present some classes of stationary and non-stationary
reducible generalized Bratteli diagrams with infinitely many odometers and
find their sets of probability ergodic invariant measures. We give examples of
diagrams that: (i) have a unique probability ergodic invariant measure, (ii)
have countably many probability ergodic invariant measures, (iii) have no
probability invariant measure, but possess an infinite $\sigma$-finite
invariant measure that takes finite values on all cylinder sets, and (iv) have
no invariant measure that takes finite values on all cylinder sets.

###  5.1. Stationary generalized Bratteli diagrams with infinitely many
odometers.

In this subsection, we describe all probability ergodic invariant measures for
a class of stationary generalized Bratteli diagrams. We apply two methods: the
construction of measure extension and the procedure of obtaining a measure
from a positive eigenvector and eigenvalue. We show that these two approaches
lead to the same ergodic invariant measures. These procedures also allow us to
obtain infinite $\sigma$-finite ergodic invariant measures.

Theorems 5.1, 5.3, and 5.5 show that for stationary reducible generalized
Bratteli diagrams, unlike the case of standard diagrams, if a class of
vertices is distinguished it doesn’t necessarily mean that the corresponding
ergodic invariant measure is finite. We will use the notation introduced in
Section 4.

######  Theorem 5.1.

Let $B$ be a stationary generalized Bratteli diagram with incidence matrix

$F=\begin{pmatrix}a&1&0&0&0&\ldots\\\ 0&a-k&1&0&0&\ldots\\\
0&0&a-k&1&0&\ldots\\\ 0&0&0&a-k&1&\ldots\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\ddots\end{pmatrix},$

where $a,k\in\mathbb{N}$ and $a-k>1$. Then there is a unique probability
ergodic invariant measure $\mu$ on $B$ if and only if $k>1$. If $k=1$ then
there are no probability invariant measures on $B$.

To prove the theorem we will need the following lemma:

######  Lemma 5.2.

For every $n\in\mathbb{N}_{0}$, we have

(5.1) $H_{i}^{(n)}=(a-(k-1))^{n}\mbox{ for }i>1.$

###### Proof of Lemma 5.2.

First notice that $H_{i}^{(n)}=H_{2}^{(n)}$ for all $n\in\mathbb{N}_{0}$ and
$i>1$. We prove the formula (5.1) by induction. For all $i>1$ we have

$H_{i}^{(0)}=1\mbox{ and }H_{i}^{(1)}=(a-k)+1.$

Assume that $H_{i}^{(n)}=(a-(k-1))^{n}$ for all $i>1$. Then

$H_{i}^{(n+1)}=(a-k)H_{i}^{(n)}+H_{i+1}^{(n)}=(a-k+1)(a-(k-1))^{n}=(a-(k-1))^{n+1}.\qed$

###### Proof of Theorem 5.1.

By Theorem 3.1 and Lemma 5.2, we have

$\widehat{\overline{\mu}}_{1}(\widehat{X}_{\overline{B}_{1}})=1+\sum_{n=0}^{\infty}\frac{H_{2}^{(n)}}{a^{n+1}}=1+\sum_{n=0}^{\infty}\frac{(a-(k-1))^{n}}{a^{n+1}}=1+\frac{1}{a}\sum_{n=0}^{\infty}\left(1-\frac{k-1}{a}\right)^{n}.$

Let $q=1-\frac{k-1}{a}\leq 1$. Assume that $k>1$, so $q<1$. Then

$\widehat{\overline{\mu}}_{1}(\widehat{X}_{\overline{B}_{1}})=1+\frac{1}{a}\sum_{n=0}^{\infty}q^{n}=1+\frac{1}{a(1-q)}=1+\frac{a}{a(k-1)}=1+\frac{1}{k-1}.$

If $k=1$, then
$\widehat{\overline{\mu}}_{1}(\widehat{X}_{\overline{B}_{1}})=\infty$.

For any $k\geq 1$ and $i>1$ we also have

$\widehat{\overline{\mu}}_{i}(\widehat{X}_{\overline{B}_{i}})=1+\sum_{n=0}^{\infty}\frac{H_{i+1}^{(n)}}{(a-k)^{n+1}}=1+\sum_{n=0}^{\infty}\frac{(a-k+1)^{n}}{(a-k)^{n+1}}=\infty.$

Thus, by Theorem 4.3, for $k=1$, there are no probability invariant measures
on $B$, and for each $k>1$, there is a unique probability ergodic invariant
measure $\mu$ on $B$ which is an extension of the unique invariant measure
from the first odometer. ∎

######  Theorem 5.3.

For $k>1$, the unique (up to constant multiple) ergodic invariant measure
$\mu$ on $B$ from Theorem 5.1 can be obtained by using the eigenvalue
$\lambda=a$, the corresponding eigenvector

(5.2)
$\xi=(\xi_{i})=\left(1,\frac{1}{k},\frac{1}{k^{2}},\ldots,\frac{1}{k^{i-1}},\ldots\right)^{T}$

and the formula

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

where $\overline{e}(w,v)$ is a finite path that begins at $w\in V_{0}$ and
ends at $v\in V_{n}$ (recall that we identify the vertices $v\in V_{n}$ with
natural numbers $i\in\mathbb{N}$).

For $k=1$, there is an ergodic infinite $\sigma$-finite invariant measure
$\mu$ which takes finite positive values on all cylinder sets. This measure
can be obtained as the extension of the unique invariant measure from the
first odometer as in Theorem 5.1 or using the eigenvalue $\lambda=a$, the
corresponding eigenvector

$\xi=(\xi_{i})=\left(1,1,1,\ldots\right)^{T}$

and formula (5.3).

###### Proof.

First we prove that, for $k\geq 1$, the vector $\xi=(\xi_{i})$ given by the
formula (5.2) is a right eigenvector for $A=F^{T}$ associated with the
eigenvalue $\lambda=a$. Indeed, for any eigenvector $\eta=(\eta_{i})$ and
eigenvalue $\lambda$, we have $a\eta_{1}=\lambda\eta_{1}$. If we take
$\eta_{1}=1$, then $\lambda=a$. From the equality
$\eta_{1}+(a-k)\eta_{2}=\lambda\eta_{2}$ we obtain $1+(a-k)\eta_{2}=a\eta_{2}$
and $\eta_{2}=\frac{1}{k}$. It is easy to prove by induction that, for every
$i\in\mathbb{N}$, we have $\eta_{i}=\frac{1}{k^{i-1}}=\xi_{i}$. Note that for
$k=1$ we obtain eigenvector $\xi=(1,1,1,\ldots)^{T}$. By Theorem 2.7, the
measure $\mu$ defined on each cylinder set by formula (5.3) determines a tail-
invariant measure on $B$ which takes finite values on cylinder sets. We show
that this measure coincides with $\widehat{\overline{\mu}}_{1}$ on all
cylinder sets. First we prove that
$\mu([{\overline{e}}^{(m)}(2)])=\widehat{\overline{\mu}}([{\overline{e}}^{(m)}(2)])$
for all $m\in\mathbb{N}$, where $[{\overline{e}}^{(m)}(2)]$ is any cylinder
set of length $m$ which ends in vertex $2$.

We have
$\mu([{\overline{e}}^{(m)}(1)])=\widehat{\overline{\mu}}_{1}([{\overline{e}}^{(m)}(1)])=\frac{1}{a^{m}}$
for all $m\in\mathbb{N}_{0}$. Then we can check that

$\widehat{\overline{\mu}}_{1}([{\overline{e}}^{(0)}(2)])=\sum_{n=0}^{\infty}\frac{(a-k)^{n}}{a^{n+1}}=\frac{1}{a}\cdot\frac{1}{1-\frac{a-k}{a}}=\frac{1}{k}=\mu([{\overline{e}}^{(0)}(2)]).$

Similarly, since the diagram is stationary, we obtain

$\widehat{\overline{\mu}}_{1}([{\overline{e}}^{(1)}(2)])=\sum_{n=0}^{\infty}\frac{(a-k)^{n}}{a^{n+2}}=\frac{1}{a^{2}}\cdot\frac{1}{1-\frac{a-k}{a}}=\frac{1}{ak}=\mu([{\overline{e}}^{(1)}(2)]).$

In general, for every $m\in\mathbb{N}_{0}$,

$\widehat{\overline{\mu}}_{1}([{\overline{e}}^{(m)}(2)])=\sum_{n=0}^{\infty}\frac{(a-k)^{n}}{a^{n+m+1}}=\frac{1}{a^{m+1}}\cdot\frac{1}{1-\frac{a-k}{a}}=\frac{1}{a^{m}k}=\mu([{\overline{e}}^{(m)}(2)]).$

Now we can use the knowledge about the measures of cylinder sets that end in
the second vertex and similarly show that

$\widehat{\overline{\mu}}_{1}([{\overline{e}}^{(0)}(3)])=\sum_{n=0}^{\infty}\frac{(a-k)^{n}}{ka^{n+1}}=\frac{1}{ak}\cdot\frac{1}{1-\frac{a-k}{a}}=\frac{1}{k^{2}}=\mu([{\overline{e}}^{(0)}(3)]).$

In general, we can prove by induction on $i$ that for every $i\in\mathbb{N}$
and $m\in\mathbb{N}_{0}$ we have

$\widehat{\overline{\mu}}_{1}([{\overline{e}}^{(m)}(i+1)])=\sum_{n=0}^{\infty}\frac{(a-k)^{n}}{k^{i-1}a^{n+m+1}}=\frac{1}{a^{m+1}k^{i-1}}\cdot\frac{1}{1-\frac{a-k}{a}}=\frac{1}{k^{i}a^{m}}=\mu([{\overline{e}}^{(m)}(i+1)]).\qed$

######  Remark 5.4.

For $i>1$ and $k\in\mathbb{N}$, the measure $\widehat{\overline{\mu}}_{i}$
does not attain finite values on all cylinder sets. In particular, for every
$i>1$ we have

$\widehat{\overline{\mu}}_{i}([{\overline{e}}^{(0)}(i+1)])=\infty.$

Indeed, we obtain

$\widehat{\overline{\mu}}_{i}([{\overline{e}}^{(0)}(i+1)])=\sum_{n=0}^{\infty}\frac{(a-k)^{n}}{(a-k)^{n+1}}=\sum_{n=0}^{\infty}\frac{1}{a-k}=\infty.$

More generally, the following theorem holds:

######  Theorem 5.5.

Let $B$ be a stationary generalized Bratteli diagram with the incidence matrix

$F=\begin{pmatrix}a_{1}&1&0&0&\ldots\\\ 0&a_{2}&1&0&\ldots\\\
0&0&a_{3}&1&\ldots\\\ 0&0&0&a_{4}&\ldots\\\
\vdots&\vdots&\vdots&\vdots&\ddots\end{pmatrix},$

where $a_{1}>a_{k}$ for all $k\geq 2$. Then there is an invariant measure
$\mu$ on $B$ generated by the right eigenvector $\mathbf{x}$ of $A=F^{T}$
associated to the eigenvalue $\lambda=a_{1}$, where

$\mathbf{x}^{T}=\left(1,\;\frac{1}{a_{1}-a_{2}},\;\frac{1}{(a_{1}-a_{2})(a_{1}-a_{3})},\;\ldots,\quad\frac{1}{(a_{1}-a_{2})(a_{1}-a_{3})\cdots(a_{1}-a_{n})},\;\ldots\;\right).$

This measure can be also obtained as an extension of the probability measure
from the odometer that passes through the diagram’s first vertex. The measure
$\mu$ is finite if and only if

$\sum_{n=2}^{\infty}\prod_{k=2}^{n}\frac{1}{a_{1}-a_{k}}<\infty.$

###### Proof.

It is straightforward to check that $\mathbf{x}^{T}F=a_{1}\mathbf{x}$. We show
that the measure $\mu$ generated by $a_{1}$ and $\mathbf{x}$ can be obtained
as an extension of the unique invariant probability measure
$\overline{\mu}_{1}$ from the first odometer.

We have

$\mu([{\overline{e}}^{(m)}(1)])=\widehat{\overline{\mu}}_{1}([{\overline{e}}^{(m)}(1)])=\frac{1}{a_{1}^{m}}$

for all $m\in\mathbb{N}_{0}$. Then for every $m\in\mathbb{N}_{0}$,

$\widehat{\overline{\mu}}_{1}([{\overline{e}}^{(m)}(2)])=\sum_{n=0}^{\infty}\frac{a_{2}^{n}}{a_{1}^{n+m+1}}=\frac{1}{a_{1}^{m+1}}\cdot\frac{1}{1-\frac{a_{2}}{a_{1}}}=\frac{1}{(a_{1}-a_{2})a_{1}^{m}}=\mu([{\overline{e}}^{(m)}(2)])$

and

$\displaystyle\widehat{\overline{\mu}}_{1}([{\overline{e}}^{(m)}(i+1)])=$
$\displaystyle\
\sum_{n=0}^{\infty}\frac{a_{i+1}^{n}}{a_{1}^{n+m+1}\prod_{j=1}^{i}(a_{1}-a_{j})}$
$\displaystyle=$ $\displaystyle\
\frac{1}{a_{1}^{m+1}\prod_{j=1}^{i}(a_{1}-a_{j})}\cdot\frac{1}{1-\frac{a_{i+1}}{a_{1}}}$
$\displaystyle=$ $\displaystyle\
\frac{1}{\prod_{j=1}^{i+1}(a_{1}-a_{j})a_{1}^{m}}=\mu([{\overline{e}}^{(m)}(i+1)]).\qed$

######  Remark 5.6.

Notice that in the case when there is $m\in\mathbb{N}$ such that $a_{m}>a_{k}$
for all $k>m$ then there is an eigenvector

$\textbf{x}^{T}=\left(0,\;\ldots,\;0,\;1,\;\frac{1}{a_{m}-a_{m+1}},\;\ldots,\;\frac{1}{(a_{m}-a_{m+1})\cdots(a_{m}-a_{m+n})},\;\ldots\;\right)$

corresponding to the eigenvalue $\lambda=a_{m}$. These eigenvalue and
eigenvector generate an invariant measure which is the extension of a
probability measure from the odometer that passes through the vertex $m$ of
the diagram.

The following example was considered in [BJKS23] and presents a stationary
generalized Bratteli diagram with infinitely many odometers such that there
does not exist any tail-invariant measure on $X_{B}$ that assigns finite
values to all cylinder sets. Indeed, it is straightforward to check that the
measure extension from every odometer is infinite, moreover, for every
$i\in\mathbb{N}$, we have

$\widehat{\overline{\mu}}_{i}([{\overline{e}}^{(0)}(i+1)])=\infty.$

######  Example 5.7.

Let $B=B(F)$ be a generalized stationary Bratteli diagram as shown in Figure 1
and given by $\mathbb{N}\times\mathbb{N}$ incidence matrix

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

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

$\ldots$$\ldots$$\ldots$$\vdots$$\vdots$$\vdots$ Figure 1. A stationary
generalized Bratteli diagram with no invariant measure that takes finite
values on all cylinder sets (illustration to Example 5.7).

###  5.2. Non-stationary generalized Bratteli diagrams with infinitely many
odometers.

Now, we modify the diagram given in Example 5.7 and consider a non-stationary
generalized Bratteli diagram $B$ defined by a sequence of natural numbers
$\\{a_{i}:i\in\mathbb{N}_{0}\\}$. Without loss of generality, we can assume
that $a_{n}\geq 2$ for all $n$. The diagram $B$ consists of an infinite
sequence of non-stationary odometers connected with the neighboring odometer
by single edges. More precisely, let $V_{n}=\mathbb{N}$ and for any
$n\in\mathbb{N}_{0}$, the incidence matrix $F_{n}$ has the form

(5.5) ${F}_{n}=\begin{pmatrix}a_{n}&1&0&0&\ldots&\ldots&\ldots&\ldots\\\
0&a_{n}&1&0&0&\ldots&\ldots&\ldots\\\ 0&0&a_{n}&1&0&0&\ldots&\ldots\\\
0&0&0&a_{n}&1&0&0&\ldots\\\
\ldots&\ldots&\ldots&\ldots&\ldots&\ldots&\ldots&\ldots\\\ \end{pmatrix}.$

Fix $i\geq 1$, set $W_{n}=\left\\{i\right\\}$, $n=1,2,\ldots$ and define the
subdiagram ${\overline{B}}_{i}=(\overline{W},{\overline{E}})$ as above with
the only difference that the set $\overline{E}_{n}$ is formed now by $a_{n}$
edges connecting the vertices $i\in V_{n}$ and $i\in V_{n+1}$. The unique tail
invariant probability measure $\overline{\mu}=\overline{\mu}_{i}$ on
${\overline{B}}_{i}=(\overline{W},{\overline{E}})$ is given by the formula
$\overline{\mu}([\overline{e}])=\dfrac{1}{a_{0}\ \cdots\ a_{n}}$ where
$r(\overline{e})=i\in V_{n+1}$.

By definition of the diagram, $H^{(n)}=H^{(n)}_{i}$ for all $n$ and $i$ (as
usual, we set $H^{(0)}_{i}=1$). Then
$H^{(n+1)}=H_{i}^{(n+1)}=(a_{n}+1)H^{(n)}$ which implies that

$H^{(n+1)}=(a_{0}+1)\ \cdots\ (a_{n}+1),\quad n\in\mathbb{N}_{0}.$

######  Theorem 5.8.

For every $i$, the extension $\widehat{\overline{\mu}}_{i}$ of
$\overline{\mu}_{i}$ is finite if and only if

$\sum_{n\geq 0}\frac{1}{a_{n}}<\infty.$

If the extensions $\widehat{\overline{\mu}}_{i}$ for $i\in\mathbb{N}$ are
finite then the set of all ergodic finite tail invariant measures on the path
space of the diagram $B$ coincides with $\\{\widehat{\overline{\mu}}_{i}\
:i\in\mathbb{N}\\}$.

###### Proof.

Using Theorem 3.1, we compute
$\widehat{\overline{\mu}}_{i}(\widehat{X}_{\overline{B}_{i}})$ as follows:

$\displaystyle\widehat{\overline{\mu}}_{i}(\widehat{X}_{\overline{B}_{i}})=$
$\displaystyle\ 1+\sum_{n\geq
0}\sum_{w\in{W}^{\prime}_{n}}{f}_{iw}^{(n)}H_{w}^{(n)}\frac{1}{a_{0}\ \cdots\
a_{n}}$ $\displaystyle=$ $\displaystyle\ 1+\sum_{n\geq
1}\frac{{(a}_{0}+1)\cdot\ldots\cdot(a_{n-1}+1)}{a_{0}\cdot\ldots\cdot a_{n}}$
$\displaystyle=$ $\displaystyle\ 1+\sum_{n\geq
0}\frac{1}{a_{n}}\cdot\left[\prod_{j=0}^{n-1}\left(1+\frac{1}{a_{j}}\right)\right]$
$\displaystyle\leq$ $\displaystyle\
1+\prod_{j=0}^{\infty}\left(1+\frac{1}{a_{j}}\right)\cdot\sum_{n=0}^{\infty}\frac{1}{a_{n}}.$

Thus, the extensions $\widehat{\overline{\mu}}_{i}$ are finite if and only if
$\sum_{n\geq 0}\frac{1}{a_{n}}<\infty.$ By Theorem 4.3, if the extensions
$\widehat{\overline{\mu}}_{i}$ are finite then the measures
$\\{\widehat{\overline{\mu}}_{i}:i\in\mathbb{N}\\}$ form the set of all
probability ergodic invariant measures on $B$. ∎

##  6\. Vershik maps on generalized Bratteli diagrams with infinitely many
odometers

In this section, we consider different orders on reducible generalized
Bratteli diagrams with infinitely many odometers (see Section 4). We present
different classes of orders on such diagrams and study whether the
corresponding Vershik map can be extended to a homeomorphism.

Recall that each vertex $v$ of (DIO) is determined by the pair $(n,i)$, where
$n\geq 0$ is the level such that $v\in V_{n}$ and $i=1,2,\ldots$ is the vertex
number of $v$ inside $V_{n}$. We will write $v=(n,i)$. For $v=(n,i)\in
V\setminus V_{0}$, we have
$r^{-1}(v)=\\{e_{1},e_{2},\ldots,e_{a_{n}^{(i)}},f\\}$, where
$e_{1},e_{2},\ldots,e_{a_{n}^{(i)}}$ are edges between $v$ and $w=(n-1,i)$,
whereas $f$ is the edge between $v$ and $w=(n-1,i+1)$.

Let $\omega=\\{\omega_{v},\;v\in V\setminus V_{0}\\}$ be an order of (DIO),
i.e., for every $v\in V\setminus V_{0}$, ${\omega}_{v}$ is a linear order on
the set $r^{-1}(v)$. Let ${\overline{e}}_{v}$ and $\widehat{e}_{v}$ be the
minimal and the maximal edges in $r^{-1}(v)$ respectively. We will say that
the order ${\omega}_{v}$ is left if ${\overline{e}}_{v}=f$, is right if
$\widehat{e}_{v}=f$ and is middle if ${\overline{e}}_{v}$ and
$\widehat{e}_{v}$ are different from $f$. According to Claim 4.1, every
maximal (minimal) infinite path
$\overline{e}=\\{{e^{\prime}}_{1},{e^{\prime}}_{2},\ldots\\}$ is contained in
a unique $\widehat{X}_{\overline{B}_{i}}$, $i\geq 1$. We will say that the
odometer $\overline{B}_{i}$ is finite-right (finite-left) if there exists a
number $n_{i}$ such that the orders ${\omega}_{v}$ are not right (not left)
for $n\geq n_{i}$, $v=(n,i)$.

Since each $\overline{B}_{i}$ is an odometer, the set
$\widehat{X}_{\overline{B}_{i}}$ consists of all paths that eventually pass
vertically through vertex $i$. Since two different minimal paths cannot pass
through the same vertex, we obtain that the set
$\widehat{X}_{\overline{B}_{i}}$ can contain not more than one infinite
minimal path. If $\widehat{X}_{\overline{B}_{i}}$ does contain an infinite
minimal path then this path is eventually vertical, hence the odometer
$\overline{B}_{i}$ should be finite-left. Similarly, the set
$\widehat{X}_{\overline{B}_{i}}$ contains the unique maximal infinite path if
and only if the odometer $\overline{B}_{i}$ is finite-right (we denote $i\in
I_{fr}$). We summarize the above remarks in the following proposition:

######  Proposition 6.1.

The set $\widehat{X}_{\overline{B}_{i}}$ contains the unique maximal (minimal)
infinite path $\widehat{e}_{i,\max}$ (${\overline{e}}_{i,\min}$) iff the
odometer $\overline{B}_{i}$ is finite-right (finite-left). For the remaining
odometers, $\widehat{X}_{\overline{B}_{i}}\cap X_{\max}=\emptyset$
($\widehat{X}_{\overline{B}_{i}}\cap X_{\min}=\emptyset$).

Let $\varphi_{\omega}\colon(X_{B}\setminus X_{\max})\rightarrow(X_{B}\setminus
X_{\min}$) be the Vershik map defined by the order $\omega$. We now consider
different orders $\omega$ on $B$ and examine when $\varphi_{\omega}$ can be
extended to a homeomorphism. Since $B$ is not stationary, strictly speaking,
we cannot consider stationary orders on $B$. We call an order on $B$ quasi-
stationary if for every vertex $i$ we fix whether it is left-ordered, right-
ordered, or middle, and the type or order depends only on the vertex number
but not on the level to which vertex belongs.

######  Theorem 6.2.

Let $I_{fr}$ ($I_{fl}$) be the set of all $i\in\mathbb{N}$ such that
$\overline{B}_{i}$ is finite-right (finite-left). Then

(i) $\varphi_{\omega}$ extends to a Borel Vershik map on $X_{B}$ if and only
if

$|I_{fr}|=|I_{fl}|;$

(ii) if

$I_{fr}=I_{fl}=\emptyset,$

then $\varphi_{\omega}$ is a homeomorphism of $X_{B}$;

(iii) for any quasi-stationary order $\omega$ on $B$, the map
$\varphi_{\omega}$ cannot extend to a Vershik homeomorphism of the whole path-
space $X_{B}$.

###### Proof.

Part (i) is obvious since equality $|I_{fr}|=|I_{fl}|$ means that
$|X_{max}|=|X_{min}|$. Since the sets $X_{max}$ and $X_{min}$ are always
closed, the condition $|X_{max}|=|X_{min}|$ is necessary and sufficient for
$\varphi_{\omega}$ to be extended to a Borel bijection of the whole space
$X_{B}$.

(ii) If $I_{fr}=I_{fl}=\emptyset$, then $\varphi_{\omega}$ is a homeomorphism
of the whole path space $X_{B}$.

(iii) Note that the case $I_{fr}=I_{fl}=\emptyset$ is not possible for a
quasi-stationary order. Indeed, if the vertex $i\in\mathbb{N}$ is right-
ordered then $i\in I_{fl}$ and there is a vertical minimal path that passes
through the vertex $i$ on each level. Similarly, if the vertex $i$ is left-
ordered then $i\in I_{fr}$ and there is a vertical maximal path passing
through the vertex $i$. If the order $\omega_{i}$ of the vertex $i$ is middle
then there are both infinite maximal path and infinite minimal path passing
through the vertex $i$. Assume that the map $\varphi_{\omega}$ can be extended
to a Vershik homeomorphism on $X_{B}$. Then we have $|I_{fr}|=|I_{fl}|$.
Moreover, since the order is quasi-stationary, we have
$|I_{fr}|=|I_{fl}|=\aleph_{0}$. We will call a cylinder set maximal (minimal)
if it corresponds to a maximal (minimal) finite path.

First, assume that $i\in I_{fr}$ and $i>1$.

Suppose $i-1\in I_{fr}\cap I_{fl}$. Then for every $n$, every maximal cylinder
set of length $n$ which ends in $i$ contains a non-maximal cylinder subset
$[\overline{g}_{1}]$ of length $n+1$ which also ends in $i$ and a non-maximal
cylinder subset $[\overline{g}_{2}]$ of length $n+1$ which ends in $i-1$. The
image $\varphi_{w}([\overline{g}_{2}])$ belongs to the minimal cylinder set of
length $n$ which passes through vertex $i-1$ on each level. The image
$\varphi_{w}([\overline{g}_{1}])$ belongs to the minimal cylinder set of
length $n$ which ends in $i$ or $i+1$ and never passes through vertex $i-1$.
Thus, we cannot extend $\varphi_{\omega}$ continuously to the maximal path
which passes through vertex $i$.

If $i-1\in I_{fl}\setminus I_{fr}$, then the maximal edge which ends in $i-1$
is not vertical. For every $n$, every maximal cylinder set of length $n$ which
passes through vertex $i$ contains a non-maximal cylinder subset
$[\overline{g}_{1}]$ of length $n+1$ which also ends in $i$ and a non-maximal
cylinder subset $[\overline{g}_{2}]$ of length $n+2$ which passes through the
maximal edge between vertices $i$ on level $n$ and $i-1$ on level $n+1$, and
then through a non-maximal edge between vertex $i-1$ on level $n+1$ and vertex
$i-1$ on level $n+2$. Again, the image $\varphi_{w}([\overline{g}_{1}])$
belongs to the minimal cylinder set of length $n$ which ends in $i$ or $i+1$,
$\varphi_{w}([\overline{g}_{2}])$ belongs to the minimal cylinder set of
length $n$ which passes through vertex $i-1$ on each level.

Thus, if $i\in I_{fr}$ and $i-1\in I_{fl}$ then $\varphi_{\omega}$ cannot be
extended to a homeomorphism. It remains to notice that such a situation will
always occur for any quasi-stationary order. Indeed, we have
$|I_{fr}|=|I_{fl}|=\aleph_{0}$, hence we have infinitely many vertical
infinite minimal paths and infinitely many vertical infinite maximal paths.
Let $j\in\mathbb{N}$ be a vertex through which passes an infinite minimal
path. Then there exists a vertex $k>j$ through which passes an infinite
maximal path. Thus, we will necessarily find a natural number $i\in(j,k]$ such
that $i\in I_{fr}$ and $i-1\in I_{fl}$. ∎

Acknowledgements.

We are very grateful to our colleagues, especially, P. Jorgensen, T. Raszeja,
S. Sanadhya, M. Wata for valuable discussions. S.B. and O.K. are also grateful
to the Nicolas Copernicus University in Toruń for its hospitality and support.
S.B. acknowledges the hospitality of AGH University of Kraków during his
visit. O.K. is supported by the NCN (National Science Center, Poland) Grant
2019/35/D/ST1/01375 and by the program ‘‘Excellence initiative - research
university’’ for the AGH University of Kraków. We thank the referees for their
detailed comments on the paper which helped us to improve the exposition.

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