# Shadows of quintessential dark energy black holes in the domain of outer
communication

Balendra Pratap Singha<EMAIL_ADDRESS>aDepartment of Applied Sciences
and Engineering, Tula’s Institute, Dehradun, Uttarakhand 248197, India

###### Abstract

The rotating black holes in the quintessential dark energy correspond to three
horizons: inner, outer, and quintessential horizon. The domain of outer
communication is the region between outer and quintessential horizon. Here, in
this work we study the photon region and shadows of the quintessential dark
energy black holes when the observer stays statically in the domain of outer
communication. The quintessential dark energy black holes shadow characterizes
by its mass $(M)$, spin parameter $(a)$, quintessential dark energy parameter
$(\omega_{q})$, and normalization factor $(\gamma)$. The dark energy parameter
$\omega_{q}$ can take values in between $-1.1<\omega_{q}<-1/3$ and follows the
equation of state $\omega_{q}$=pressure$(p)$/energy density($\rho_{q})$. This
state parameter significantly affects the shape and size of the black hole
shadow. We generalize all the geodesic equations of motion for $\omega_{q}$
and obtain relation to visualize the black hole shadow by a static observer at
any arbitrary distance in the domain of outer communication. We analytically
estimate the black hole shadow observables: radius $R_{s}$, distortion
parameter $\delta_{s}$ and the shadow area $A$. Using the numerical values of
shadow radius $R_{s}$ and area $A$, we obtain the angular diameter of the
black hole shadow. The angular size of the M87 and Sgr A∗ black holes are $\
42\pm 3\mu as$ and $48.7\pm 7\mu as$ respectively as observe by Event Horizon
Telescope (EHT). In this case, the angular diameter of the black hole shadow
increases with the quintessence parameter $\omega_{q}$ and takes values
$\theta_{d}\approx 20\pm 3{{}^{o}}$ with the parameter
$-0.66\leq\omega_{q}\leq-0.62$ for the static observer at $r_{o}=5M$ in the
domain of outer communication.

##  I Introduction

The observations of type Ia supernovae (SNe Ia) lie between the red shift
range $0.16\leq z\leq 0.62$, confirming that our Universe is going under the
late time acceleration SupernovaSearchTeam:1998fmf ;
SupernovaCosmologyProject:1998vns . According to general relativity, this
cosmic acceleration indicates that there exists some strange energy component
in the Universe which is called dark energy. The observational constrains over
the state parameter $\omega_{q}$ provided by the large state structure of the
Universe and the cosmic microwave background (CMB) is $-1.1<\omega_{q}<-1/3$
suzuki:2012 ; Planck:2015bue ; Chiba:2012cb . The hypothesis of dark energy is
compatible with the standard model of big bang cosmology ($\Lambda$CDM model)
when the dark energy state parameter is exactly equal to -1. For this value of
state parameter, the dark energy is considered as cosmological constant, which
is interestingly agreements with the observations but still there is a
possibility that some significant component of the dark energy densities have
state parameters other than -1 Caldwell:1997ii . One of the simplest is the
quintessence dark energy model in which the dynamical scalar field is
minimally coupled with the gravity Caldwell:1997ii . The quintessence dark
energy is dynamic and time-varying which is different from the cosmological
constant model which does not change with time. Some researchers also consider
the quintessence dark energy as the fifth fundamental force responsible for
the expansion of the Universe Carroll:1998zi ; Cicoli:2012tz ; Dvali:2001dd .

The asymptotic structure of the black hole gets modified in the presence of
the quintessential dark energy. The black hole spacetime remains no more
asymptotically flat in quintessence due to the cosmological horizon. The very
first model of the black hole in quintessence was presented by Kislev
Kiselev:2002dx . After that, several researchers intensively studied the
properties of spherically symmetric black holes in quintessence dark energy
ChenWang:2008 ; Ding:2013 ; Fernando:2012 ; Wang:2009 ; Jamil:2015 ;
Kalam:2014 ; Xiao:2010 ; Ghaderi:2015a ; Ghaderi:2015b ; Biswas:2015 ;
Thomas:2012 ; Uniyal:2015 ; Kuriakose:2009 ; Kar:2006 ; Pandey:2014ona ;
Priyabrat:2015 ; Singh:2014a ; Singh:2014b ; Xu:2016ylr ; Xu:2017vse . The
Lovelock black holes in quintessence have been studied by Ghosh:2017cuq ;
Toledo:2019 . The study of Narnia black holes in quintessence has been done by
Fernando:2013mex . The authors of Pedraza:2020uuy  discussed the geodesics of
the Hayward black hole in quintessence. Thermodynamics of the Bardeen black
hole in quintessential dark energy studied in Wu:2022fkz . The rotating
counterpart of the spherically symmetric black hole in quintessence obtained
by Toshmatov:2015npp  and Ghosh:2015ovj . The study of rotating anti-de-sitter
and rotating charged anti-de-sitter black holes in the presence of perfect
fluid matter have been intensively studied by Xu:2017bpz  and Xu:2017vse . The
author of Belhaj:2020rdb  extended Schwarzschild black holes in quintessence
up to $D$-dimensional spacetimes and studied gravitational lensing and shadow
properties.

The observational results from the Event Horizon Telescope (EHT) proved that
black holes are just not only theoretical concepts. The first picture of the
Messier 87 (M87) black hole revealed by the EHT group in 2019 Akiyama:2019cqa
; Akiyama:2019brx ; Akiyama:2019sww ; Akiyama:2019bqs ; Akiyama:2019fyp ;
Akiyama:2019eap . Black holes are completely dark objects but interestingly
they cast a shadow Goddi:2016qax . The incoming photons towards the black hole
horizon which have quite large angular momentum fall inside the black hole
event horizon and create a dark spot, and photons that carries a little bit
smaller angular momentum form the photon region around the black hole
Goddi:2016qax . This dark spot surrounded with the bright photon rings is
called a black hole shadow Perlick:2021aok . Casting shadow by the black hole
also verifies the existence of the event horizon Akiyama:2019cqa . Recently
EHT collaborators published the image of Sgr A∗ black hole shadow. This
supermassive black hole lies at the heart of our galaxy Milky way
Akiyama:2022L12 ; Akiyama:2022L13 ; Akiyama:2022L14 ; Akiyama:2022L15 ;
Akiyama:2022L16 ; Akiyama:2022L17 . The first analytical study of this subject
was done by Synge Synge:1966  and Luminet Luminet:1979 . Later Bardeen
extended it for the rotating spacetime Bardeen1 . In the past few years,
several researchers analytically studied the black hole shadow for rotating
and non-rotating spacetimes.Amarilla:2010zq ; Amarilla:2013sj ; Yumoto:2012kz
; Abdujabbarov:2016hnw ; Amir:2016cen ; Tsukamoto:2014tja ; Bambi:2010hf ;
Takahashi:2005hy ; Wei:2013kza ; Abdujabbarov:2012bn ; Amarilla:2011fx ;
Bambi:2008jg ; Atamurotov:2013sca ; Wang:2017hjl ; Schee:2008kz ;
Grenzebach:2014fha ; Abdujabbarov:2015xqa ; Cunha:2015yba ; Cunha:2016bpi ;
Singh:2017xle ; Singh:2022dqs ; Kumar:2017tdw ; Kumar:2019ohr . The optical
properties of the rotating and non-rotating black holes in the presence of the
plasma medium have been studied by Perlick:2015vta ; Atamurotov:2015nra ;
Abdujabbarov:2015pqp  This subject has been extended for the higher
dimensional spacetime by Papnoi:2014aaa ; Abdujabbarov:2015rqa ; Singh:2017vfr
; Amir:2017slq ; Belhaj:2022cdh . Various researchers estimated black hole
parameters using the properties of the black hole shadow KumarWalia:2022ddq ;
KumarWalia:2022aop ; Vagnozzi:2022moj ; Ghosh:2022jfi ; Afrin:2021imp .

The primary goal of this work is to analytically study the angular size of the
photon regions for rotating black holes in quintessential dark energy in the
domain of outer communication. The rotating black hole in quintessence
corresponds to three horizons: inner, outer, and quintessential or
cosmological event horizons Toshmatov:2015npp . The domain of outer
communication is the region where the observer statically stays in between the
outer and the cosmological horizon Perlick:2021aok . The quintessence field
affects the shadow region, and the appearance of the bright photons region
varies as seen by the observer. The size of the black hole shadow decreases
and gets distorted with the state parameter $\omega_{q}$. The photon region
will appear deviated and more close to the event horizon in the presence of
the quintessential dark energy.

This paper is organized as follows: In Sec. II, we discuss the rotating and
non-rotating black hole metric with the quintessential dark energy. In Sec.
III, we derive the effective potential of the black hole and find the impact
parameters. We analytically study the black hole shadow in Sec. IV and for
better understating of the effect of quintessential dark energy on the black
hole shadow, we determine the shadow observables and angular size in Sec. V.
Finally, we conclude our results in Sec. VI.

##  II  Black Hole Metric

###  II.1 Non-rotating black hole metric in the presence of quintessential
dark energy

---
Figure 1:  Plot showing the horizons of the non-rotating black holes .

The spherically symmetric black hole metric with quintessential dark energy
has been derived by Kislev Kiselev:2002dx . In Boyer Lindquist coordinates and
natural units $G=c=1$, the black hole metric is given by

$\displaystyle
ds^{2}=-f(r)dt^{2}+f^{-1}(r)dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}),$ (1)

with

$\displaystyle f(r)=1-\frac{2M}{r}-\frac{\gamma}{r^{3\omega_{q}+1}},$ (2)

where $M$ is the black hole, $\gamma$ is the normalization factor and
$\omega_{q}$ is the quintessential state parameter. In Fig. 1, we plot the
metric function $f(r)$ with radial distance $r$ for different values of
$\omega_{q}$ and $\gamma$. For $\omega_{q}=-2/3$, the quintessential dark
energy significantly affects the black hole horizons. Non-degenerate and
degenerate horizons exist for the corresponding values of normalization
parameter $\gamma$. If we consider $\omega_{q}=-1/3$, the black hole metric
reduces for the Kottler spacetime also degenerate and non-degenerate horizons
exist with the corresponding values of $\gamma$. For $\omega_{q}=-1$, the
black hole metric corresponds single horizon (cf. Table 1 and Fig. 1). In the
absence of the quintessential field the black hole metric (1) reduces for the
Schwarzschild spacetime.

$\omega_{q}=-2/3$ | $\omega_{q}=-1/3$ | $\omega_{q}=-1$ |
---|---|---|---
$\gamma$ | Inner Horizon | Outer Horizon | $\gamma$ | Inner Horizon | Outer Horizon | $\gamma$ | Horizon |
0.030 | no horizon | no horizon | 0.100 | 2.763932 | 7.236067 | 0.080 | 2.173913 |
0.037 | 2.946526 | 3.056142 | 0.120 | 4.000000 | 4.000000 | 0.900 | 2.197802 |
0.045 | 2.752125 | 2.752125 | 0.150 | no horizon | no horizon | 0.100 | 2.222222 |
Table 1: The horizons of non-rotating black holes in quintessential dark
energy

###  II.2  Rotating black holes in quintessential dark energy

---
Figure 2:  Plot showing the horizons of the rotating black hole in
quintessential dark energy. $\omega_{q}=-2/3$
---
$a$ | $\gamma$ |  | Inner
---
Horizon
| Outer
---
Horizon
| Quintessential
---
Horizon
0.70 | 0.13 | 0.283718 | 3.590570 | 3.590570
0.70 | 0.11 | 0.394402 | 2.309770 | 6.386736
1.07 | 0.11 | 1.263181 | 1.263181 | 6.564545
1.15 | 0.10 | No Horizon | No Horizon | 7.598965
Table 2: Table showing the numerical values for the horizons of the rotating
black hole in quintessential dark energy

The rotating black hole metric surrounded with the quintessential dark energy
is

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-\left(1-\frac{2Mr+\gamma
r^{1-3\omega_{q}}}{\Sigma}\right)dt^{2}+\frac{\Sigma}{\Delta}dr^{2}-2a\sin^{2}\theta\left(\frac{2Mr+\gamma
r^{1-3\omega_{q}}}{\Sigma}\right)d\phi dt+\Sigma d\theta^{2}$ (3)
$\displaystyle+\sin^{2}\theta\left[r^{2}+a^{2}+a^{2}\sin^{2}\theta\left(\frac{2Mr+\gamma
r^{1-3\omega_{q}}}{\Sigma}\right)\right]d\phi^{2}\ ,$

where

$\displaystyle\Delta(r)=r^{2}-2Mr+a^{2}-\gamma r^{1-3\omega_{q}},$ (4)

and

$\displaystyle\Sigma=r^{2}+a^{2}\cos^{2}\theta\ .$ (5)

We get the horizons of the black hole by simply solving $\Delta{(r)}=0$. For
$\omega_{q}=-2/3$, the equation (4) reduces to the cubic polynomial equation,
and we find the radius of the horizon by solving

$r^{2}-2Mr+a^{2}-\gamma r^{3}=0,$ (6)

which also can be written as

$\gamma(r-r_{in})(r-r_{out})(r-r_{q})=0,$ (7)

where $r_{in}$ and $r_{out}$ are the inner and outer horizon, and $r_{q}$
corresponds to the quintessential horizon respectively. The rotating black
hole in the quintessential dark energy corresponds to three horizons which
vary with the spin parameter $a$ and the normalization parameter $\gamma$ as
shown in Table (2). In Fig. 2, we show the variations of horizons with the
corresponding values of $a$ and $\gamma$. For extremely rotating case
$a=1.15$, the black hole corresponds only quintessential horizon (cf. Fig. 2
and Table (2)).

##  III Effective Potential and Impact Parameters

---
Figure 3:  Plot showing the variation of effective potential with $r$ for
different values of $a$, $\omega_{q}$ and $\gamma$.
---
Figure 4:  Plot showing the variation of critical radius $r_{cric}$ with $\gamma$ for different values of quintessential parameter $\omega_{q}$ and spin parameter $a$. $a=0.9$ | $a=0.7$
---|---
$\omega_{q}=-2/3$ | $\omega_{q}=-1/3$ | $\omega_{q}=-2/3$ | $\omega_{q}=-1/3$
$\gamma$ | $r_{cric}$ | $\gamma$ | $r_{cric}$ | $\gamma$ | $r_{cric}$ | $\gamma$ | $r_{cric}$
0.08 | 2.51 | 0.08 | 2.57 | 0.08 | 2.57 | 0.08 | 2.63
0.10 | 2.45 | 0.10 | 2.54 | 0.10 | 2.54 | 0.10 | 2.60
0.12 | 2.42 | 0.12 | 2.48 | 0.12 | 2.48 | 0.12 | 2.54
0.14 | 2.36 | 0.14 | 2.45 | 0.14 | 2.45 | 0.14 | 2.51
Table 3: Table showing the variation of $r_{cric}$ with $a$, $\omega_{q}$ and
$\gamma$

The motion of particles and electromagnetic radiations around the rotating
spacetime is regular so the equations of motion of a particle around a black
hole in quintessential dark energy are completely integrable. In this section,
we derive complete integral equations of motion and study the effective
potential of the black hole and calculate the impact parameters. We use the
Hamilton-Jacobi formalism to find the complete geodesic equations which were
originally derived by Carter wheeler ; novikov ; Chandrasekhar:1992 ;
Carter:1968rr . The Hamilton-Jacobi equation reads

$\displaystyle\frac{\partial
S}{\partial\upsilon}=-\frac{1}{2}g^{\mu\nu}p_{\mu}p_{\nu},$ (8)

where $\upsilon$ is the affine parameter and $p_{\mu}$ is the conjugate
momenta corresponding to the first derivative of Jacobian action $S$ with
respect to the generalized coordinate. We choose a separable solution for the
Jacobean action in the given form

$\displaystyle S=\frac{1}{2}{m_{0}}^{2}\upsilon-{\cal E}t+{\cal
L}\phi+S_{r}(r)+S_{\theta}(\theta),$ (9)

where $m_{0}$ is the rest mass of the test particle which is zero for the
photon. The black hole metric is cyclic for time $t$ and azimuthal $\phi$
coordinates. There exist two conserved quantities corresponding to these
cyclic coordinates $\cal{E}$ and $\cal{L}$ having units of energy and angular
momentum respectively. Using our separable solution defined in Eq. (9), we
obtain our complete equations of motion for a massive test particle in the
first-order differential form as

$\displaystyle\Sigma\frac{dt}{d\upsilon}$ $\displaystyle=$
$\displaystyle\frac{r^{2}+a^{2}}{\Delta(r)}\left[{\cal E}(r^{2}+a^{2})-a{\cal
L}\right]-a(a{\cal E}\sin^{2}\theta-{\mathcal{L}})\ ,$ (10)
$\displaystyle\Sigma\frac{dr}{d\upsilon}$ $\displaystyle=$
$\displaystyle\sqrt{\mathcal{R}(r)}\ ,$ (11)
$\displaystyle\Sigma\frac{d\theta}{d\upsilon}$ $\displaystyle=$
$\displaystyle\sqrt{\Theta(\theta)}\ ,$ (12)
$\displaystyle\Sigma\frac{d\phi}{d\upsilon}$ $\displaystyle=$
$\displaystyle\frac{a}{\Delta(r)}\left[{\cal E}(r^{2}+a^{2})-a{\cal
L}\right]-\left(a{\cal E}-\frac{{\cal L}}{\sin^{2}\theta}\right)\ ,$ (13)

where

$\displaystyle\mathcal{R}(r)=\left[(r^{2}+a^{2}){\cal E}-a{\cal
L}\right]^{2}-\Delta(r)\left[{m_{0}}^{2}r^{2}+(a{\cal E}-{\cal L})^{2}+{\cal
K}\right],$ (14) $\displaystyle\Theta(\theta)={\cal K}-\left[\frac{{\cal
L}^{2}}{\sin^{2}\theta}-a^{2}{\cal E}^{2}\right]\cos^{2}\theta\ .$ (15)

---
Figure 5:  Plot showing the shadows of the rotating black holes in
quintessential dark energy for the different values of the parameter $\gamma$,
$a$.
---
Figure 6:  Plot showing the shadows of the rotating black holes in
quintessential dark energy for the different values of the parameter
$\omega_{q}$, $a$ .

Here, $\mathcal{K}$ is the Carter separable constant. The dynamics of a test
particle around the black hole can be obtained from the geodesic Eqs.
(10)-(13). These geodesic equations reduce for the Kerr black holes in the
absence of the quintessential field $\gamma=0$ novikov . The study of the
motion of the test particle around the rotating black hole requires two impact
parameters. Here, we are introducing two impact parameters $\eta$ and $\xi$ in
terms of conserved quantities $\mathcal{E}$, $\mathcal{L}$ and $\mathcal{K}$
as

$\xi=\mathcal{L}/\mathcal{E},\quad\quad\eta=\mathcal{K}/\mathcal{E}^{2}.$ (16)

The rest mass of the photon is zero, so for the case of null geodesics, we
consider $m_{0}=0$. The radial equation of motion in terms of impact
parameters reads

$\mathcal{R}(r)=\frac{1}{{\cal{E}}^{2}}\left[[(r^{2}+a^{2})-a{\xi}]^{2}-\Delta(r)[(a-{\xi})^{2}+{\eta}]\right].$
(17)

---
Figure 7:  Plot showing the variation of shadow radius $R_{s}$ with the
quintessential dark energy parameter $\omega_{q}$ for different $\gamma$.

The formation of circular orbits around the black hole is an important feature
for the study of gravitational lensing and shadow properties. The idea of the
formation of circular orbits can be obtained from the effective potential of
the black hole. The circular photon orbits may be categorized into two types:
unstable or stable circular photon orbits. When a ray of photon departs from
the unstable circular orbit then it may fall into the black hole or reach the
infinite observer. No bound orbits exit around the unstable circular orbits.
Instead of that, many possible bound orbits may exist around the stable
circular orbits. We use the radial equation of motion (17) to find the
expression of the effective potential. The radial equation of motion can be
rewritten in terms of effective potential as $V_{eff}$ as

$\left(\frac{d{r}}{d\upsilon}\right)^{2}+V_{eff}(r)=0,$ (18)

from the above equation (18), one can find the expression of the effective
potential which takes the following form

$V_{eff}=\frac{1}{{{\Sigma}^{2}}}[((r^{2}+a^{2})-a{\xi})^{2}-\Delta(r)((a-{\xi})^{2}+{\eta})].$
(19)

In Fig. (3), we plot the behaviour of effective potential with radial
coordinate $r$ for different values of $a$, $\omega_{q}$ and $\gamma$. One can
observe from Fig. (3) that for a fixed value of $a$ and $\omega_{q}$ the
maximum value or peak value increases with the increasing value of $\gamma$.
Another interesting result one can observe is that the critical radius
corresponding to the peak value of effective potential shifts to the left
which means the radii of the unstable circular orbits decreases with the
increasing values of $\gamma$ (cf. Fig. (4) and Table 3). The critical radius
$r_{cri}$ of unstable photon orbits corresponds to the local maximum of the
effective potential while the critical radius for the most stable photon
orbits corresponds to the local minimum. Here we are interested to find the
most unstable photon orbits and for that, we must have to maximize the
effective potential which satisfies the following condition

$V_{eff}=\frac{\partial V_{eff}}{\partial
r}=0\;\;\;\;\mbox{or}\;\;\;\mathcal{R}=\frac{\partial\mathcal{R}}{\partial
r}=0,$ (20)

by applying the above conditions (20), we obtain the individual expressions
for impact parameters

$\displaystyle\eta$ $\displaystyle=$
$\displaystyle\frac{1}{a^{2}\left(-3\gamma\omega_{q}+\gamma+2(M-r)r^{3\omega_{q}}\right){}^{2}}\Big{[}r^{3}(-4\left(r(r-3M)^{2}-4a^{2}M\right)r^{6\omega_{q}})$
$\displaystyle+$ $\displaystyle 4\gamma
r^{3\omega_{q}}\left(3\omega_{q}\left(2a^{2}+r(r-3M)\right)+2a^{2}+3r(r-3M)-9{\gamma}^{2}r\left(\omega_{q}+1\right){}^{2}\right)\Big{]},$
$\displaystyle\xi$ $\displaystyle=$
$\displaystyle\frac{1}{a\left(-3\gamma\omega_{q}+\gamma+2(M-r)r^{3\omega_{q}}\right)}\Big{[}2\left(a^{2}(M+r)+r^{2}(r-3M)\right)r^{3\omega_{q}}$
(22) $\displaystyle+$
$\displaystyle\gamma\left(-3\left(a^{2}+r^{2}\right)\omega_{q}+a^{2}-3r^{2}\right)\Big{]}.$

---
Figure 8:  Plot showing the variation of the distortion parameter with the
quintessential dark energy parameter $\omega_{q}$ for different parameter
$\gamma$ .

These expressions of $\eta$ and $\xi$ define the motion of photons around the
black holes in quintessential dark energy. These equations of impact
parameters exactly reduce for the Kerr black holes in the absence of the
quintessence dark energy field novikov .

##  IV Black hole shadow

A black hole form shadow due to the infalling light rays inside its event
horizon. The light rays at the unstable equilibrium form orbit around it. We
consider orthogonal tetrad to define the celestial coordinate along the
boundary of the black hole shadow. The orthonormal tetrads are

$\displaystyle
e_{0}=\frac{(r^{2}+a^{2})\partial_{t}+a\partial_{\phi}}{\sqrt{\Sigma\Delta}},\;\;\;\;e_{1}=\frac{1}{\sqrt{\Sigma}}\partial_{\theta},$
$\displaystyle
e_{2}=-\frac{\partial_{\phi}+a\sin^{2}{\theta}\partial_{t}}{\sqrt{\Sigma\Delta}},\;\;\;\;e_{3}=-\sqrt{\frac{\Delta}{\Sigma}}\partial_{r}.$
(23)

The tangent vector over a light ray can be defined as

$\dot{\lambda}(\upsilon)=\dot{t}\partial_{t}+\dot{r}\partial_{r}+\dot{\theta}\partial_{\theta}+\dot{\phi}\partial_{\phi},$
(24)

and the tangent vector for the observers position is

$\dot{\lambda}(\upsilon)=\alpha(-e_{0}+\sin\Phi\cos\Psi e_{1}+\sin\Phi\sin\Psi
e_{2}+\cos\Phi e_{3}).$ (25)

The expression for the scalar factor $\alpha$ can be determined by comparing
Eq. (24) and (25), and using Eq. (10 \- 13), which takes the following form

$\alpha=\frac{a\mathcal{L}-(r^{2}+a^{2})\mathcal{E}}{\sqrt{\Sigma\Delta}}.$
(26)

Next, the celestial coordinates $\phi$ and $\psi$ can be defined as

$\displaystyle\sin\Psi(r_{O},r_{p})$ $\displaystyle=$
$\displaystyle\left.\left(\frac{\xi-a}{\sqrt{(a-\xi)^{2}+\eta}}\right)\right|_{r=r_{O}},$
(27) $\displaystyle\sin\Phi(r_{O},r_{p})$ $\displaystyle=$
$\displaystyle\left.\left(\frac{\sqrt{\Delta[(a-\xi)^{2}+\eta]}}{(r^{2}+a^{2}-a\xi)}\right)\right|_{r=r_{O}},$
(28)

where the impact parameters $\xi$ and $\eta$ are functions of the radius of
the unstable photon orbits around the black hole. To visualize a shadow, we
introduce the Cartesian coordinates $X$ and $Y$ Grenzebach:2014fha

$\displaystyle
X(r_{O},r_{p})=-2\tan\left(\frac{\Phi(r_{O},r_{p})}{2}\right)\sin(\Psi(r_{O},r_{p})),$
$\displaystyle
Y(r_{O},r_{p})=-2\tan\left(\frac{\Phi(r_{O},r_{p})}{2}\right)\cos(\Psi(r_{O},r_{p})),$
(29)

which satisfies the following relation

$X^{2}+Y^{2}=4\tan^{2}\Big{(}\frac{\Phi(r_{o},r_{p})}{2}\Big{)}.$ (30)

The contour plot of $X$ vs $Y$ defines the photon region or shadow boundary of
rotating black hole in quintessential dark energy as seen by static observer
at the distance $r_{O}=5M$ in the domain of outer communication. We plot
various figures for the black hole shadow in quintessence for different values
of spin parameter $a$, state parameter $\omega_{q}$ and normalization factor
$\gamma$ (cf. Fig. 5 and 6). The effective size of the black hole shadow
decreases with $\omega_{q}$ (Fig. 5) while the shadow size increases for the
increasing values of $\gamma$ (Fig. 6). The next important result one can
observe from the shadow plots, the shape of the black hole shadow distorted
with the quintessence parameter $\omega_{q}$ while for the increasing values
of normalization factor $\gamma$ the black hole shadow appears more circular
even for the very high values of spin parameter which we have clearly shown in
the first plot of the figure 5.

$\omega_{q}$ | $\gamma=0.09$ | $\gamma=0.10$
---|---|---
$R_{s}$ | $\delta_{s}$ | $A$ | $\theta_{d}$ | $R_{s}$ | $\delta_{s}$ | $A$ | $\theta_{d}$
-0.66 | 0.9376 | 0.07086 | 2.7617000 | 21.489792 | 0.8988 | 0.05900 | 2.53790873 | 20.600496
-0.65 | 0.9662 | 0.07469 | 2.93281007 | 22.145304 | 0.9388 | 0.06100 | 2.76882836 | 21.517296
-0.64 | 0.9894 | 0.08221 | 3.07534388 | 22.677048 | 0.9734 | 0.06754 | 2.97668279 | 22.310328
-0.63 | 1.0090 | 0.08504 | 3.19839579 | 23.12628 | 0.9985 | 0.07036 | 3.13217494 | 22.88562
-0.62 | 1.0250 | 0.09492 | 3.30063578 | 23.49301 | 1.0250 | 0.07374 | 3.30063578 | 23.49300
$\omega_{q}$ | $\gamma=0.11$ | $\gamma=0.12$
$R_{s}$ | $\delta_{s}$ | $A$ | $\theta_{d}$ | $R_{s}$ | $\delta_{s}$ | $A$ | $\theta_{d}$
-0.66 | 0.8359 | 0.08500 | 2.1951213 | 19.158828 | 0.7055 | 0.03883 | 1.5636657 | 16.17006
-0.65 | 0.8959 | 0.09100 | 2.52155791 | 20.534028 | 0.8194 | 0.04637 | 2.1093167 | 18.780648
-0.64 | 0.9460 | 0.09700 | 2.81146153 | 21.68232 | 0.8973 | 0.05215 | 2.52944481 | 20.566116
-0.63 | 0.9877 | 0.09900 | 3.06478477 | 22.638084 | 0.9584 | 0.06103 | 2.88564886 | 21.966528
-0.62 | 1.0170 | 0.1000 | 3.24931472 | 23.30964 | 1.0010 | 0.06463 | 3.14787898 | 22.942920
Table 4:  Table Showing the black hole shadow observables: shadow radius
$R_{s}$, distortion $\delta_{s}$, Area $A$ and angular diameter $\theta_{d}$
for spin parameter $a=1.0$ with different $\omega_{q}$ and $\gamma$.

##  V Observables

Figure 9:  Plot showing the variation of the black hole shadow area $A$ with
the quintessential dark energy parameter $\omega_{q}$

In this section we estimate observables: shadow radius $R_{s}$, distortion
parameter $\delta_{S}$ respectively, which is originally shared by Hioki and
Maeda in Hioki:2009na . The observable shadow radius $R_{s}$ is given by

$R_{s}=\frac{(X_{t}-X_{r})^{2}+Y_{t}^{2}}{2|X_{r}-X_{t}|},$ (31)

where $(X_{t},Y_{t})$, $(X_{r},Y_{r})$ are the top most and right most co-
ordinates of the $X$ and $Y$ axis in the Fig. 5 and 6. Another observable
which is distortion parameter reads in the form

$\delta=\frac{D}{R_{s}},$ (32)

where $D=|X_{l}-X_{l}^{{}^{\prime}}|$ with $X_{l}$ and $X_{l}^{{}^{\prime}}$
are the left most co-ordinates on the $X$ axis from where the black hole
shadow and reference circle passes.

In Fig. 7 and 8, we plot the variation of shadow radius $R_{s}$ and distortion
parameter with the quintessence parameter $\omega_{q}$ for different values of
normalization factor $\gamma$. The shadow radius $R_{s}$ increases with the
$\omega_{q}$ while it decreases for the increasing $\gamma$ that we have shown
in the Fig. 7. The distortion parameter increases with $\omega_{q}$ (cf. Fig.
8). The black hole shadow appears more distorted in the presence of
quintessential field.

The angular diameter $\theta_{d}$ of the black hole shadow can be estimated
via

$\theta_{d}=\frac{2}{r_{o}}\times\sqrt{\frac{A}{\pi}},$ (33)

where $A$ is the area of the black hole shadow in quintessential dark energy.
The variation of area of the black hole shadow $A$ with the quintessence
parameter $\omega_{q}$ have shown in the Fig. (9) and Table 4. The shadow area
$A$ monotonically increase with the parameter $\omega_{q}$ while it decreases
with the parameter $\gamma$ (cf. Fig. 9 and Table 4). Using the Eq. (33), we
analytically estimate the angular diameter of the black hole shadow in
quintessential dark energy. The angular diameter of the black hole shadow
varies from $\theta_{d}\approx 21^{o}$ to $23^{o}$ with the variation of
$\omega_{q}$ from $-0.66$ to $-0.62$ for the parameter $\gamma=0.09$. Our
results shows that the angular diameter of the black hole shadow monotonically
increases with the quintessence dark energy parameter $\omega_{q}$. As we
further increase the parameter $\gamma$ from $0.09$ to $0.12$, the angular
diameter of the black hole shadow decreases, for $\gamma=0.12$ it varies from
$\theta_{d}\approx 16^{o}$ to $22^{o}$ as clearly shown in the Fig. 10 and
Table 4.

Figure 10:  Plot showing the variation of angular diameter of the black hole
shadow with the quintessential dark energy parameter $\omega_{q}$.

##  VI Conclusion

The observations from EHT show that the angular size of the M87∗ black hole
shadow is $42\pm 3\mu as$, and very recent results confirm the size of the
supermassive black hole Sgr∗ $48.7\pm 7\mu as$. These astrophysical objects
are very far from the earth so the observed size of these black hole’s shadow
are very small such as 16.4 $megaparsec$ for M87 and 7.86 $kiloparsec$ for Sgr
A∗. Here, we have analytically estimated the angular size of the
quintessential dark energy black holes in the domain of outer communication.
First, we derive the null geodesic equations of motion and obtained the
effective potential for the black hole. After maximizing the effective
potential we find impact parameters. We consider our observer in between the
outer and cosmological horizon so we introduced corresponding celestial
coordinates and plotted various figures of black hole shadow for different
values of quintessential dark energy parameter $\omega_{q}$, normalization
factor $\gamma$ and spin parameter $a$. For the analytical study of the shape
and size of the black hole shadow image, we numerically studied the shadow
observables. In our study, we have found the size of the black hole shadow
increases with the quintessence dark energy parameter $\omega_{q}$ while it
decreases with the normalization factor $\gamma$. Next, the shape of the
shadow, which is observable by the distortion parameter $\delta_{s}$,
increases with the parameter $\omega_{q}$ while it decreases with the
parameter $\gamma$. The observable shadow area $A$ also corresponds to the
same results as the shadow radius and using the numerical results of the
shadow area, we estimated the angular diameter of the quintessence dark energy
black holes in the domain of outer communication. The angular diameter of the
black hole shadow increases with the quintessence parameter $\omega_{q}$ and
varies from $\theta_{d}\approx 16{{}^{o}}$ to 23o for
$-0.66\leq\omega_{q}\leq-0.62$ and $0.09\leq\gamma\leq 0.12$ with spin
parameter $a=1.0$ when the observer stays at rest in between the outer and the
cosmological horizon of the quintessential dark energy black holes.

##  VII Acknowledgement

The author would like to thank Rahul Kumar Walia for the fruitful discussion
and Tula’s Institute for providing research facilities.

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# On the stability and shadowing of tree-shifts of finite type

Dawid Bucki<EMAIL_ADDRESS>
###### Abstract.

We investigate relations between the pseudo-orbit-tracing property,
topological stability and openness for tree-shifts. We prove that a tree-shift
is of finite type if and only if it has the pseudo-orbit-tracing property
which implies that the tree-shift is topologically stable and all shift maps
are open. We also present an example of a tree-shift for which all shift maps
are open but which is not of finite type. It also turns out, that if a
topologically stable tree-shift does not have isolated points then it is of
finite type.

###### 2020 Mathematics Subject Classification:

Primary 37B10, 37B51 Secondary 37B25, 37B65

##  1\. Introduction

Tree-shifts were introduced and studied in [1, 2, 3, 4, 5] as a step-in
between one-dimensional shifts and multi-dimensional shifts defined over
$\mathbb{Z}^{d}$. They are less complicated than multi-dimensional shifts but
still enrich the dynamics of the classical one-dimensional case. At the same
time they preserve a lot of important properties of one-dimensional shifts,
which often fail for the general $\mathbb{Z}^{d}$-case. The dynamics of tree-
shifts is further studied in [6] and [7] from which we derive our basics
notions and terminology.
Let $\Sigma$ be some finite non-empty set. A tree is a function assigning a
label from some finite alphabet $\mathcal{A}$ to every node of the infinite
$|\Sigma|$-ary tree which can be indexed by words over the alphabet $\Sigma$.
Without loss of generality, one may consider only labeled binary trees, as all
proofs easily generalise from the binary case to the case of $\Sigma$ with
more elements. The space of all trees can be endowed with a metric compatible
with the product topology on $\mathcal{A}^{\Sigma^{*}}$, where $\Sigma^{*}$ is
the set of all words over $\Sigma$ endowed with the discrete topology. There
are naturally arising shift maps denoted by $\sigma^{i}$, where the index $i$
is in $\Sigma$. The $i$-th map applied to a tree, gives us a sub-tree rooted
at $i$-th node and relabels the nodes accordingly. A tree-shift is then
defined as a closed subset of the family of all labeled trees invariant with
respect to all shift actions $\sigma^{i}$, $(i\in\Sigma)$.
It turns out that tree-shifts can be characterised by the forbidden sets of
blocks, where a block is a labeled finite tree. This is analogous to the way
one-dimensional shifts are defined. For every tree-shift there is some set of
blocks such that a labeled tree is in the tree-shift if and only if none of
the forbidden blocks is a sub-block of the tree. If for a tree-shift $X$ this
forbidden set defining $X$ can be finite then we say that $X$ is a tree-shift
of finite type.
The pseudo-orbit-tracing property (sometimes also called the shadowing
property) and topological stability were introduced in [8] and [15]
respectively. They play a key role in the theory of stability of dynamical
systems. The pseudo-orbit-tracing property was also considered for actions of
other groups and semi-groups in [10] and in [13]. In [15] Walters proved that
one-dimensional shifts are of finite type if and only if they have the pseudo-
orbit-tracing property. In [12], Oprocha showed that the same is true for
multi-dimensional shifts over $\mathbb{Z}^{d}$. In section 3 we generalise
Walter’s proof obtaining similar result for tree-shifts. In section 4 we
define topological stability for tree-shifts in an analogous way to [15] and
prove that if a tree-shift is of finite type, then it is topologically stable.
We prove that topologically stable tree-shifts without isolated points have
the pseudo-orbit-tracing property, therefore being of finite type. This result
was proved for zero-dimensional spaces equipped with a single transformation
in [9] by Kawaguchi.
In section 5 we consider a generalisation of the notion of open shifts. This
was introduced for one-dimensional shift spaces by Parry in [14], where it is
proved that the shift is open if and only if it is of finite type. We prove
that for tree-shifts, being of finite type implies that all shift maps are
open and provide an example showing that the converse is false. By doing so we
provide another example showing that not all classical theorems concerning
one-dimensional shifts of finite type can be generalised to tree-shifts (for
another example of this type, see Theorem 3.3 in [6]). To the best of the
author’s knowledge there are no results characterising openness of shifts over
semigroups. Most of the multi-dimensional symbolic dynamics concerns actions
of countable groups, where by definition every transformation in the action is
a homeomorphism hence it is open.

This paper arose from author’s Bachelor’s Thesis written under the supervision
of Dominik Kwietniak. In the final stages of this research, the author was
supported by National Science Centre (NCN), Poland grant no.
2022/47/O/ST1/03299. For the purpose of Open Access, the author has applied a
CC-BY public copyright licence to any Author Accepted Manuscript (AAM) version
arising from this submission. The author would like to express his gratitude
to D. Kwietniak for his advices, introducing the author to the problem and for
general guidance in author’s first years of mathematical education.

The author also wants to thank anonymous reviewers for their careful reading
of the manuscript and their comments that improved the quality of this
article.

##  2\. Preliminaries

By $\mathbb{N}$ we understand the set of non-negative integers and $n$ always
denotes an integer. Fix some alphabet (non-empty finite set) $\Sigma$. For
$n\geq 0$, $\Sigma^{n}$ stands for the set of words of length $n$ over the
alphabet $\Sigma$ with $\Sigma^{0}=\\{\epsilon\\}$, where $\epsilon$ is the
empty word. By $\Sigma^{*}=\bigcup_{n\geq 0}\Sigma^{n}$ we denote the set of
all words over the alphabet $\Sigma$ and
$\Sigma^{<n}=\bigcup_{k=0}^{n-1}\Sigma^{k}$ denotes the set of words whose
length is smaller than $n\in\mathbb{N}$. For the word $w$ over $\Sigma$ and
the set $B\subseteq\Sigma^{*}$ we put $wB=\\{wu\ |\ u\in B\\}$, where $wu$ is
the concatenation of the words $w$ and $u$. For alphabets $\Sigma$ and
$\mathcal{A}$, a labeled tree on $\Sigma$ over $\mathcal{A}$ is a function
$t\colon\Sigma^{*}\to\mathcal{A}$. One can imagine the set $\Sigma^{*}$ as an
unlabeled $|\Sigma|$-ary infinite tree whose nodes correspond to words in
$\Sigma^{*}$ and $\epsilon$ is the root. Then labeled tree $t$ assigns to
every node $w\in\Sigma^{*}$ a letter $t_{w}\in\mathcal{A}$. On the space
$\mathcal{T}$ of all labeled trees on $\Sigma$ over $\mathcal{A}$, we define a
metric $d$, by putting for labeled trees $s$ and $t$,

($*$) $d(s,t)=\begin{cases}2^{-n-1},&\text{if }n=\min\\{|w|\ |\ s_{w}\neq
t_{w}\\}<\infty,\\\ 0,&\text{otherwise},\end{cases}$

where $|w|$ is the length of the word $w$. Observe that the metric space
$(\mathcal{T},d)$ is compact.
From now on, we will refer to labeled trees as trees for simplicity.

######  Remark 2.1.

Note that for trees $s,t\in\mathcal{T}$ and $n\in\mathbb{N}$ we have that
$d(s,t)<2^{-n}$ if and only if
${\left.\mkern-1.0mus\right|}_{\Sigma^{<n}}={\left.\mkern-1.0mut\right|}_{\Sigma^{<n}}$.

For two words $u,w\in\Sigma^{*}$, we say that $u$ is a prefix of $w$ if there
is some word $v\in\Sigma^{*}$, such that $uv=w$. We say that a set
$C\subseteq\Sigma^{*}$ is prefix-closed if for any $w\in C$ and $u$ being a
prefix of $w$, we have $u\in C$. For example $\Sigma^{<n}$ is prefix-closed
for every $n\in\mathbb{N}$. Finite prefix-closed sets $C\subseteq\Sigma^{*}$
correspond to finite subtrees of the full $|\Sigma|$-ary tree rooted at the
vertex $\epsilon$. Given a finite prefix-closed set $C\subseteq\Sigma^{*}$, a
function $p\colon C\to\mathcal{A}$ is called a pattern on $C$ over
$\mathcal{A}$. Patterns correspond to labelled finite subtrees. A pattern
$p\colon C\to\mathcal{A}$ is a sub-pattern of a tree $t\in\mathcal{T}$ if
there is a word $w\in\Sigma^{*}$ such that

${\left.\mkern-1.0mut\right|}_{wC}=p.$

If there is no such word $w$ for a pattern $p$, then we say that the tree $t$
omits $p$. Patterns with domain $\Sigma^{<n}$, for $n>0$, are called blocks of
height $n$.

To deal with dynamics on trees, we introduce the shift maps
$\sigma^{i}\colon\mathcal{T}\to\mathcal{T}$, for $i\in\Sigma$, by defining
$\sigma^{i}(t)_{w}=t_{iw}$ for $t\in\mathcal{T}$. To shorten the notation we
define $\sigma^{w}=\sigma^{w_{n-1}}\circ\ldots\circ\sigma^{w_{0}}$ for
$w=w_{0}\ldots w_{n-1}\in\Sigma^{n}$. Then $\sigma^{w}(t)_{u}=t_{wu}$ for
every $u,v\in\Sigma^{*}$ and $t\in\mathcal{T}$. We also agree that
$\sigma^{\epsilon}=\text{id}$.

######  Definition 2.2.

A set $X\subseteq\mathcal{T}$ is called a tree-shift (on $\Sigma$ over
$\mathcal{A}$) if $X$ is closed and for every $i\in\Sigma$ it holds
$\sigma^{i}(X)\subseteq X$.

We can observe that if $\Sigma$ is a single element, then Definition 2.2
yields a classical one-sided shift over the alphabet $\mathcal{A}$.

For a block $b$ of height $n$ and a tree-shift $X$ we define the cylinder of
$b$ as

$[b]=\left\\{t\in X\ |\
{\left.\mkern-1.0mut\right|}_{\Sigma^{<n}}=b\right\\}.$

Note that $[b]$ depends on the choice of $X$ but we supress that from our
notation. We also denote the set of all blocks of height $n\geq 0$ over
$\mathcal{A}$, appearing in trees from a tree-shift $X$, as
$\mathcal{B}_{n}(X)$ and put

$\mathcal{B}(X)=\bigcup_{n>0}\mathcal{B}_{n}(X).$

The set $\left\\{[b]\ |\ b\in\mathcal{B}(X)\right\\}$ is then a base for the
topology on $X$ given by the metric defined by ($*$ ‣ 2) as if
$b\colon\Sigma^{<n}\to\mathcal{A}$ is a block of height $n>0$ with
$b\in\mathcal{B}_{n}(X)$, then there is $t\in X$ such that
${\left.\mkern-1.0mut\right|}_{\Sigma^{<n}}=b$ and $[b]$ is the open ball of
radius $2^{-n}$ centered at $t$.

Given a family of patterns $\mathcal{F}$, we define
$\mathcal{T}_{\mathcal{F}}$ to be the set of all trees omitting all patterns
from $\mathcal{F}$. Then $\mathcal{T}_{\mathcal{F}}$ is a closed subset of
$\mathcal{T}$ and for every $i\in\Sigma$, we have
$\sigma^{i}(\mathcal{T}_{\mathcal{F}})\subseteq\mathcal{T}_{\mathcal{F}}$.
Therefore the family $\mathcal{T}_{\mathcal{F}}$ is a tree-shift. Of course
for every non-empty tree-shift $X$ we have
$X=\mathcal{T}_{\mathcal{B}(\mathcal{T})\setminus\mathcal{B}(X)}$, so every
tree-shift can be characterised by some family of forbidden patterns. Note
that the family of forbidden patterns defining $X$ is not unique (compare with
[11]).

######  Definition 2.3.

A tree-shift $X$ on $\Sigma$ over $\mathcal{A}$ is of finite type if
$X=\mathcal{T}_{\mathcal{F}}$ for some finite set of patterns $\mathcal{F}$ on
$\Sigma$ over $\mathcal{A}$.

For the tree-shift of finite type we can always assume that its forbidden set
of patterns $\mathcal{F}$ consists of blocks of some common height $n>0$,
because it is always possible to extend forbidden patterns to forbidden
patterns of arbitrary length. From now on we will be using this assumption.
For further reference we state a simple characterisation of tree-shifts of
finite type.

######  Lemma 2.4.

A tree-shift $X$ is a tree-shift of finite type if and only if there is some
$n>0$, such that for $t\in\mathcal{T}$, we have that $t\in X$ if and only if
$t|_{w\Sigma^{<n}}\in\mathcal{B}_{n}(X)$ for every $w\in\Sigma^{*}$.

##  3\. Pseudo-orbit-tracing property

In this section we generalise Theorem 1 from [15] to tree-shifts, adapting the
approach from [15] to our setting. Let $X$ be a tree-shift on $\Sigma$ over
$\mathcal{A}$.

######  Definition 3.1.

A $\delta$-pseudo-orbit in a tree-shift $X$ is a family
$\left\\{t^{(w)}\right\\}_{w\in\Sigma^{*}}$ of trees from $X$ such that for
every $w\in\Sigma^{*}$ and for any $i\in\Sigma$, we have

$d\left(\sigma^{i}\left(t^{(w)}\right),t^{(wi)}\right)<\delta.$

If $\Sigma$ has a single element then the Definition 3 reduces to the usual
definition of a $\delta$-pseudo-orbit.

We can work with $\delta$ equal $2^{-n}$ for some $n\geq 0$, since we consider
the metric $d$ on $X$ that can only attain such values. Therefore we will
write “$[n]$-pseudo-orbit”, instead of “$2^{-n}$-pseudo-orbit”. By Remark 2.1,
$\left\\{t^{(w)}\right\\}_{w\in\Sigma^{*}}$ is an $[n]$-pseudo-orbit if and
only if for every $i\in\Sigma$ and $w\in\Sigma^{*}$ we have

${\left.\mkern-1.0mu\sigma^{i}\left(t^{(w)}\right)\right|}_{\Sigma^{<n}}={\left.\mkern-1.0mut^{(w)}\right|}_{i\Sigma^{<n}}={\left.\mkern-1.0mut^{(wi)}\right|}_{\Sigma^{<n}}.$

######  Lemma 3.2.

If $n>0$ and $\left\\{t^{(w)}\right\\}_{w\in\Sigma^{*}}$ is an $[n]$-pseudo-
orbit in $X$, then for any $w,u,v\in\Sigma^{*}$, such that $uv\in\Sigma^{<n},$
we have

$t^{(w)}_{uv}=t^{(wu)}_{v}.$

###### Proof.

We will prove this lemma, by induction on the length of $u$.
If $u=\epsilon$, then of course

$t^{(w)}_{uv}=t^{(w)}_{v}=t^{(wu)}_{v}.$

Now, assume that $i\in\Sigma$ and $u\in\Sigma^{*}$ is such that
$iuv\in\Sigma^{<n}$ and

$t^{(x)}_{uv}=t^{(xu)}_{v}$

for any $x\in\Sigma^{*}$. Then

$t^{(wiu)}_{v}=t^{(wi)}_{uv}.$

But $\\{t^{(w)}\\}_{w\in\Sigma^{*}}$ is an $[n]$-pseudo-orbit and
$uv\in\Sigma^{<n}$, so

$t^{(wiu)}_{v}=t^{(wi)}_{uv}=t^{(w)}_{iuv}.\qed$

######  Definition 3.3.

Let $T=\left\\{t^{(w)}\right\\}_{w\in\Sigma^{*}}$ be a family of trees in a
tree-shift $X$, indexed by $\Sigma^{*}$. For $t\in X$ and $\varepsilon>0$, we
say that $t$ $\varepsilon$-traces the family $T$, if for any $w\in\Sigma^{*}$,
we have

$d\left(\sigma^{w}(t),t^{(w)}\right)<\varepsilon.$

Again, instead of writing “$2^{-m}$-traces” for some $m>0$, we will say
“$[m]$-traces”. If a tree $t$ $[m]$-traces a family of trees
$\left\\{t^{(w)}\right\\}_{w\in\Sigma^{*}}$, then by Remark 2.1 we have

${\left.\mkern-1.0mu\sigma^{w}(t)\right|}_{\Sigma^{<m}}={\left.\mkern-1.0mut\right|}_{w\Sigma^{<m}}={\left.\mkern-1.0mut^{(w)}\right|}_{\Sigma^{<m}}.$

######  Lemma 3.4.

If trees $r,t\in X$ $[m]$-trace
$\left\\{t^{(w)}\right\\}_{w\in\Sigma^{*}}\subseteq X$ for some $m>0$, then
$r=t$.

###### Proof.

For every $w\in\Sigma^{*}$ we have

$r_{w}=\sigma^{w}(r)_{\epsilon}=t^{(w)}_{\epsilon}=\sigma^{w}(t)_{\epsilon}=t_{w}.\qed$

######  Definition 3.5.

A tree-shift $X$ has the pseudo-orbit-tracing property, if for any $m>0$,
there is $n>0$, such that every $[n]$-pseudo-orbit in $X$ is $[m]$-traced by
some tree from $X$.

It turns out that pseudo-orbit-tracing characterises the tree-shifts of finite
type.

######  Theorem 3.6.

A tree-shift $X$ is of finite type if and only if it has the pseudo-orbit-
tracing property.

###### Proof.

We adapt the proof presented in [15]. Assume that $X$ is of finite type and
$X=\mathcal{T}_{\mathcal{F}}$ with
$\mathcal{F}\subseteq\mathcal{B}_{p}(\mathcal{T})$ for some $p>0$. Let $m>0$.
We pick $n\geq\max\\{p,m\\}$. We will prove, that every $[n]$-pseudo-orbit in
$X$ is $[m]$-traced by some point in $X$.
Let $\left\\{t^{(w)}\right\\}_{w\in\Sigma^{*}}$ be an $[n]$-pseudo-orbit in
$X$. We define $t\in\mathcal{T}$ by putting $t_{w}=t^{(w)}_{\epsilon}$ for
$w\in\Sigma^{*}$. By Lemma 3.2 for $u\in\Sigma^{<m}$ and $w\in\Sigma^{*}$, we
have

$\sigma^{w}(t)_{u}=t_{wu}=t^{(wu)}_{\epsilon}=t^{(w)}_{u}.$

Therefore

${\left.\mkern-1.0mu\sigma^{w}(t)\right|}_{\Sigma^{<m}}={\left.\mkern-1.0mut^{(w)}\right|}_{\Sigma^{<m}},$

which means that $t$ indeed $[m]$-traces
$\left\\{t^{(w)}\right\\}_{w\in\Sigma^{*}}$.

It remains to prove that $t\in X$. By the assumption that all forbidden words
have length $p$, it suffices to check that
${\left.\mkern-1.0mut\right|}_{w\Sigma^{<p}}\in\mathcal{B}_{p}(X)$, for any
$w\in\Sigma^{*}$. Fix $w\in\Sigma^{*}$. For $u\in\Sigma^{<p}$ we have
$u\in\Sigma^{<n}$ because $n\geq p$ and
$t_{wu}=t^{(wu)}_{\epsilon}=t^{(w)}_{u}$, where the latter equality follows
from Lemma 3.2. It means that for any $w\in\Sigma^{*}$ we have

${\left.\mkern-1.0mut\right|}_{w\Sigma^{<p}}={\left.\mkern-1.0mut^{(w)}\right|}_{\Sigma^{<p}}$

and since we know that

${\left.\mkern-1.0mut^{(w)}\right|}_{\Sigma^{<p}}\in\mathcal{B}_{p}(X),$

we conclude
${\left.\mkern-1.0mut\right|}_{w\Sigma^{<p}}\in\mathcal{B}_{p}(X).$

Now assume that $X$ has the pseudo-orbit-tracing property. Then there is some
$n>1$ such that every $[n-1]$-pseudo-orbit is $[1]$-traced by some point from
$X$. We will prove that a tree $t$ is in $X$ if and only if for any
$w\in\Sigma^{*}$, we have
${\left.\mkern-1.0mut\right|}_{w\Sigma^{<n}}\in\mathcal{B}_{n}(X)$. Then Lemma
2.4 will yield that $X$ is of finite type. The implication $(\Rightarrow)$ is
by the definition, so let $t$ be such that for any $w\in\Sigma^{*}$, we have
${\left.\mkern-1.0mut\right|}_{w\Sigma^{<n}}\in\mathcal{B}_{n}(X)$. This means
that for any $w\in\Sigma^{*}$, there is some $t^{(w)}\in X$, such that
${\left.\mkern-1.0mut^{(w)}\right|}_{\Sigma^{<n}}={\left.\mkern-1.0mut\right|}_{w\Sigma^{<n}}$.
Observe that the family $\left\\{t^{(w)}\right\\}_{w\in\Sigma^{*}}$ is an
$[n-1]$-pseudo-orbit in $X$, because for $i\in\Sigma$ and $w\in\Sigma^{*}$, we
have

${\left.\mkern-1.0mut^{(wi)}\right|}_{\Sigma^{<n-1}}={\left.\mkern-1.0mut\right|}_{wi\Sigma^{<n-1}}={\left.\mkern-1.0mut^{(w)}\right|}_{i\Sigma^{<n-1}}={\left.\mkern-1.0mu\sigma^{i}\left(t^{(w)}\right)\right|}_{\Sigma^{<n-1}}.$

This $[n-1]$-pseudo-orbit is then $[1]$-traced by some $r\in X$, because of
the pseudo-orbit-tracing property. It means that for every $w\in\Sigma^{*}$ it
holds

$r_{w}=\sigma^{w}(r)_{\epsilon}=t^{(w)}_{\epsilon}=t_{w}.$

So $r=t$ and thus $t\in X$.∎

##  4\. Topological stability of tree-shifts

In the theory of one-dimensional shifts it is known, that if the shift has the
pseudo-orbit-tracing property then it is topologically stable, which is proved
in [15]. Also, if a one-dimensional shift is stable and has no isolated points
then it has the pseudo-orbit-tracing property. This can be concluded from [9],
where Kawaguchi proves this fact for general zero-dimensional spaces. In this
section we extend this result to tree-shifts.

######  Definition 4.1.

A tree-shift $X$ is topologically stable if for every $\varepsilon>0$ there
exists $\delta>0$ such that if $\left\\{\tau^{i}\right\\}_{i\in\Sigma}$ is a
family of continuous maps from $X$ to itself, satisfying
$d\left(\tau^{i}(t),\sigma^{i}(t)\right)<\delta$, for any $i\in\Sigma$ and
$t\in X$, then there exists a continuous mapping $\varphi\colon X\to X$ such
that for any $i\in\Sigma$ and $t\in X$ it holds
$d\left(\varphi(t),t\right)<\varepsilon$ and
$\sigma^{i}\circ\varphi=\varphi\circ\tau^{i}$.

When we consider the set $\\{\tau^{i}\\}_{i\in\Sigma}$ as above, then for
$w\in\Sigma^{n}$, $\tau^{w}$ stands for
$\tau^{w_{n-1}}\circ\ldots\circ\tau^{w_{0}}$. Again, to make our reasoning
easier to follow, we will restate the above definition in an equivalent way.

######  Definition 4.2.

If $X$ is a tree-shift, then maps $\zeta,\xi\colon X\to X$ are said to be
$[n]$-close for $n>0$, if for any $t\in X$ we have

${\left.\mkern-1.0mu\zeta(t)\right|}_{\Sigma^{<n}}={\left.\mkern-1.0mu\xi(t)\right|}_{\Sigma^{<n}}.$

The families $\left\\{\zeta^{i}\right\\}_{i\in\Sigma}$,
$\left\\{\xi^{i}\right\\}_{i\in\Sigma}$ of the maps from $X$ to $X$ are said
to be $[n]$-close if $\zeta^{i}$ and $\xi^{i}$ are $[n]$-close for any
$i\in\Sigma$.

The following proposition follows immediately from Definitions 4.1 and 4.2.

######  Proposition 4.3.

A tree-shift $X$ is topologically stable if and only if for any $m>0$ there
exists $n>0$ such that for any family $\left\\{\tau^{i}\right\\}_{i\in\Sigma}$
of continuous mappings from $X$ to $X$ that is $[n]$-close to the family
$\left\\{\sigma^{i}\right\\}_{i\in\Sigma}$ there is a continuous map
$\varphi\colon X\to X$ that is $[m]$-close to identity and satisfies
$\varphi\circ\tau^{i}=\sigma^{i}\circ\varphi$ for any $i\in\Sigma$.

It is convenient to consider finite pseudo-orbits in a tree-shift.

######  Definition 4.4.

Let $m,N>0$. A finite $[m]$-pseudo-orbit of order $N$ in a tree shift $X$ is a
family $\left\\{t^{(w)}\right\\}_{w\in\Sigma^{<N}}$ of trees from $X$
satisfying

${\left.\mkern-1.0mu\sigma^{i}\left(t^{(w)}\right)\right|}_{\Sigma^{<m}}={\left.\mkern-1.0mut^{(wi)}\right|}_{\Sigma^{<m}},$

for every $w\in\Sigma^{<N-1}$.

For a finite pseudo-orbit we can define tracing and pseudo-orbit-tracing
property in the same way as in the infinite case.

######  Definition 4.5.

Let $n,N>0$. We say that a tree $t\in X$ $[n]$-traces a family
$\\{t^{(w)}\\}_{w\in\Sigma^{<N}}$ of trees from $X$, if for any
$w\in\Sigma^{<N}$ we have

${\left.\mkern-1.0mu\sigma^{w}(t)\right|}_{\Sigma^{<n}}={\left.\mkern-1.0mut^{(w)}\right|}_{\Sigma^{<n}}.$

######  Definition 4.6.

A tree-shift $X$ has the finite pseudo-orbit-tracing property if for any $m>0$
there is some $n>0$ such that for any $N>0$, any finite $[n]$-pseudo-orbit of
order $N$ from $X$ is $[m]$-traced by a tree from $X$.

Of course, the pseudo-orbit-tracing property implies the finite pseudo-orbit-
tracing property. In compact spaces the converse is also true.

######  Proposition 4.7.

If a tree-shift $X$ has the finite pseudo-orbit-tracing property then it has
the pseudo-orbit-tracing property.

###### Proof.

Fix $m>0$ and pick $n>0$ using the finite pseudo-orbit-tracing property. Let
$T=\left\\{t^{(w)}\right\\}_{w\in\Sigma^{*}}$ be an $[n]$-pseudo-orbit. For
$N\in\mathbb{N}$, let $t^{N}$ be a tree that $[m]$-traces the finite pseudo-
orbit $\left\\{t^{(w)}\right\\}_{w\in\Sigma^{<N}}$. Then, by compactness we
can assume that the sequence $\left\\{t^{N}\right\\}_{N>0}$ converges to some
$t\in X$ as $N\to\infty$. Let us prove that $t$ $[m]$-traces the $[n]$-pseudo-
orbit $T$. Fix $w\in\Sigma^{*}$ and take $N>|w|$ big enough to have

${\left.\mkern-1.0mu\sigma^{w}(t)\right|}_{\Sigma^{<m}}={\left.\mkern-1.0mu\sigma^{w}\left(t^{N}\right)\right|}_{\Sigma^{<m}}.$

The tree $t^{N}$ $[m]$-traces the finite pseudo-orbit
$\left\\{t^{(v)}\right\\}_{v\in\Sigma^{<N}}$ so

${\left.\mkern-1.0mu\sigma^{w}\left(t^{N}\right)\right|}_{\Sigma^{<m}}={\left.\mkern-1.0mut^{(w)}\right|}_{\Sigma^{<m}}.$

Thus we obtain that

${\left.\mkern-1.0mu\sigma^{w}(t)\right|}_{\Sigma^{<m}}={\left.\mkern-1.0mut^{(w)}\right|}_{\Sigma^{<m}}$

so the tree $t$ indeed $[m]$-traces the $[n]$-pseudo-orbit $T$. ∎

We will now prove that every tree-shift $X$ of finite type is topologically
stable. As we already know, tree-shifts of finite type have the pseudo-orbit-
tracing property. We can use this fact to obtain stability.

######  Theorem 4.8.

If a tree-shift $X$ has the pseudo-orbit-tracing property, then it is
topologically stable.

###### Proof.

We will use an equivalent condition for topological stability given by
Proposition 4.3.
Fix $m>0$. By the pseudo-orbit-tracing property there is some $n>0$ such that
every $[n]$-pseudo-orbit in $X$ is $[m]$-traced by some tree from $X$. Let
$\left\\{\tau^{i}\right\\}_{i\in\Sigma}$ be a set of mappings such that for
any $i\in\Sigma$ and $t\in X$ we have

${\left.\mkern-1.0mu\tau^{i}(t)\right|}_{\Sigma^{<n}}={\left.\mkern-1.0mu\sigma^{i}(t)\right|}_{\Sigma^{<n}}.$

Let us now construct a map $\varphi\colon X\to X$ as follows. Observe that for
every tree $t\in X$ the set $\left\\{\tau^{w}(t)\right\\}_{w\in\Sigma^{*}}$ is
an $[n]$-pseudo-orbit in $X$ as for every $i\in\Sigma$ and $w\in\Sigma^{*}$ we
have

${\left.\mkern-1.0mu\sigma^{i}\left(\tau^{w}(t)\right)\right|}_{\Sigma^{<n}}={\left.\mkern-1.0mu\tau^{i}\left(\tau^{w}(t)\right)\right|}_{\Sigma^{<n}}={\left.\mkern-1.0mu\tau^{wi}(t)\right|}_{\Sigma^{<n}}.$

This pseudo-orbit is then $[m]$-traced by some tree $\varphi(t)\in X$. By
Lemma 3.4, a tree tracing a pseudo-orbit is unique so the map
$t\mapsto\varphi(t)$ is well-defined. We now need to prove that the map
$\varphi$ is continuous. If $\left\\{t^{k}\right\\}_{k=0}^{\infty}$ is a
sequence in $X$ converging to some $t\in X$ and $\varphi\left(t^{k}\right)$ is
a tree $[m]$-tracing the $[n]$-pseudo-orbit
$\left\\{\tau^{w}\left(t^{k}\right)\right\\}_{w\in\Sigma^{*}}$ then we want to
prove that $\varphi\left(t^{k}\right)\to\varphi(t)$ as $k\to\infty$. Observe
that $\varphi\left(t^{k}\right)$ is defined by

(4.1)
$\varphi\left(t^{k}\right)_{w}=\sigma^{w}\left(\varphi\left(t^{k}\right)\right)_{\epsilon}=\tau^{w}\left(t^{k}\right)_{\epsilon}.$

It follows that for every $k$ big enough (depending on $n$),
$\varphi\left(t^{k}\right)_{w}$ is equal to $\tau^{w}(t)_{\epsilon}$ because
$\tau^{w}$ is a continuous mapping and $t^{k}$ converges. We also know that
for every $w\in\Sigma^{*}$ it holds

(4.2)
$\varphi(t)_{w}=\sigma^{w}\left(\varphi(t)\right)_{\epsilon}=\tau^{w}(t)_{\epsilon}.$

Equations (4.1) and (4.2) imply that for $w\in\Sigma^{*}$ we have

$\varphi\left(t^{k}\right)_{w}=\tau^{w}\left(t^{k}\right)_{\epsilon}\longrightarrow\tau^{w}(t)_{\epsilon}=\varphi(t)_{w}\text{
as }k\to\infty.$

It follows that $\varphi(t^{k})\to\varphi(t)$ as $k\to\infty$. The condition
of $\varphi$ being close to the identity is also satisfied as for every $t\in
X$ we have that $\varphi(t)$ $[m]$-traces the pseudo-orbit
$\left\\{\tau^{w}(t)\right\\}_{w\in\Sigma^{*}}$ so

${\left.\mkern-1.0mu\varphi(t)\right|}_{\Sigma^{<m}}={\left.\mkern-1.0mu\sigma^{\epsilon}(\varphi(t))\right|}_{\Sigma^{<m}}={\left.\mkern-1.0mu\tau^{\epsilon}(t)\right|}_{\Sigma^{<m}}={\left.\mkern-1.0mut\right|}_{\Sigma^{<m}}.$

Finally, for any $w\in\Sigma^{*}$ and $i\in\Sigma$, using (4.1), we also have

$\varphi\left(\tau^{i}(t)\right)_{w}=\sigma^{w}\left(\varphi\left(\tau^{i}(t)\right)\right)_{\epsilon}=\tau^{w}\left(\tau^{i}(t)\right)_{\epsilon}=\tau^{iw}(t)_{\epsilon}=\varphi(t)_{iw}=\sigma^{i}\left(\varphi(t)\right)_{w},$

so

$\sigma^{i}\circ\varphi=\varphi\circ\tau^{i}.$

We have constructed the mapping $\varphi$ satisfying the conditions from
Proposition 4.3, thus the tree-shift $X$ is topologically stable. ∎

######  Corollary 4.9.

Every tree-shift of finite type is topologically stable.

###### Proof.

If $X$ is a tree-shift of finite type then by Theorem 3.6, $X$ has the pseudo-
orbit-tracing property. We conclude from Theorem 4.8 that it is topologically
stable. ∎

######  Theorem 4.10.

If $X$ is a topologically stable tree-shift with no isolated points then it
has the finite pseudo-orbit-tracing property.

###### Proof.

Let $m>0$ and take $n>m$ from the definition of topological stability and let
$N>1$ (for $N\leq 1$ there is nothing to prove). We will prove that every
finite $[n]$-pseudo-orbit of order $N$ in $X$ is $[m]$-traced by some point
from $X$. Fix $T=\left\\{t^{(w)}\right\\}_{w\in\Sigma^{<N}}$ such that for
every $w\in\Sigma^{<N-1}$ and $i\in\Sigma$ we have

${\left.\mkern-1.0mu\sigma^{i}\left(t^{(w)}\right)\right|}_{\Sigma^{<n}}={\left.\mkern-1.0mut^{(wi)}\right|}_{\Sigma^{<n}}.$

The first step is to turn the map $w\mapsto t^{(w)}$ into an injection on
$\Sigma^{<N}$. Assume that there are some $u,v\in\Sigma^{<N}$ such that $u\neq
v$ and $t^{(u)}=t^{(v)}$ appear in $T$. We know that $t^{(v)}$ is not an
isolated point in $X$ (as we assumed that $X$ has no isolated points), so
there are infinitely many $t\in X$ such that

$t|_{\Sigma^{<n+1}}=t^{(v)}|_{\Sigma^{<n+1}}.$

We then choose $t$ that does not appear in
$\left\\{t^{(w)}\right\\}_{w\in\Sigma^{<N}}$ and replace $t^{(v)}$ by $t$.
After finitely many steps we obtain the family
$S=\left\\{s^{(w)}\right\\}_{w\in\Sigma^{<N}}$ such that $w\mapsto s^{(w)}$ is
an injection over $\Sigma^{<N}$ and for each $w\in\Sigma^{<N}$ we have

${\left.\mkern-1.0mus^{(w)}\right|}_{\Sigma^{<n+1}}={\left.\mkern-1.0mut^{(w)}\right|}_{\Sigma^{<n+1}}.$

Observe that $S$ is still an $[n]$-pseudo-orbit of order $N$ as for
$w\in\Sigma^{<N-1}$ we have

${\left.\mkern-1.0mu\sigma^{i}\left(s^{(w)}\right)\right|}_{\Sigma^{<n}}={\left.\mkern-1.0mus^{(w)}\right|}_{i\Sigma^{<n}}={\left.\mkern-1.0mut^{(w)}\right|}_{i\Sigma^{<n}}={\left.\mkern-1.0mu\sigma^{i}(t^{(w)})\right|}_{\Sigma^{<n}}={\left.\mkern-1.0mut^{(wi)}\right|}_{\Sigma^{<n}}={\left.\mkern-1.0mus^{(wi)}\right|}_{\Sigma^{<n}}.$

The second step is to use $S$ to construct a family of maps
$\left\\{\tau^{i}\right\\}_{i\in\Sigma}$. Our family $S$ is finite and two
trees in $S$ corresponding to different words are different, so we can find an
integer $M>\max\\{m,n\\}$ such that for any $u,v\in\Sigma^{<N}$ with $u\neq v$
we have

${\left.\mkern-1.0mus^{(u)}\right|}_{\Sigma^{<M}}\neq{\left.\mkern-1.0mus^{(v)}\right|}_{\Sigma^{<M}}.$

For $i\in\Sigma$ and $t\in X$, we define $\tau^{i}(t)$ as follows. If there is
$w\in\Sigma^{<N-1}$ such that

${\left.\mkern-1.0mut\right|}_{\Sigma^{<M}}={\left.\mkern-1.0mus^{(w)}\right|}_{\Sigma^{<M}},$

then we put $\tau^{i}(t)=s^{(wi)}.$ Observe that if such a word exists then it
is unique, because of the way we constructed $S$. Otherwise we put
$\tau^{i}(t)=\sigma^{i}(t).$ The maps $\left\\{\tau^{i}\right\\}_{i\in\Sigma}$
are well defined and it is trivial to check that they are continuous. Let us
prove that $\tau^{i}$ is $[n]$-close to $\sigma^{i}$ for every $i\in\Sigma$.
If $t\in X$ and for every $w\in\Sigma^{<N-1}$ we have

${\left.\mkern-1.0mut\right|}_{\Sigma^{<M}}\neq{\left.\mkern-1.0mus^{(w)}\right|}_{\Sigma^{<M}},$

then of course

${\left.\mkern-1.0mu\tau^{i}(t)\right|}_{\Sigma^{<n}}={\left.\mkern-1.0mu\sigma^{i}(t)\right|}_{\Sigma^{<n}}.$

If for some $w\in\Sigma^{<N-1}$ it holds

${\left.\mkern-1.0mut\right|}_{\Sigma^{<M}}={\left.\mkern-1.0mus^{(w)}\right|}_{\Sigma^{<M}},$

then

${\left.\mkern-1.0mu\tau^{i}(t)\right|}_{\Sigma^{<n}}={\left.\mkern-1.0mus^{(wi)}\right|}_{\Sigma^{<n}}={\left.\mkern-1.0mu\sigma^{i}(s^{(w)})\right|}_{\Sigma^{<n}}={\left.\mkern-1.0mus^{(w)}\right|}_{i\Sigma^{<n}}={\left.\mkern-1.0mut\right|}_{i\Sigma^{<n}}={\left.\mkern-1.0mu\sigma^{i}(t)\right|}_{\Sigma^{<n}}.$

Hence, for each $i\in\Sigma$ the map $\tau^{i}$ is $[n]$-close to $\sigma^{i}$
and by topological stability we get, that there is some continuous
$\varphi\colon X\to X$ such that

$\sigma^{i}\circ\varphi=\varphi\circ\tau^{i}$

for any $i\in\Sigma$, and $\varphi$ is $[m]$-close to the identity on $X$. We
will prove, that $\varphi\left(t^{(\epsilon)}\right)$ $[m]$-traces the
$N$-finite $[n]$-pseudo-orbit $\left\\{t^{(w)}\right\\}_{w\in\Sigma^{<n}}$.
Indeed, for $w\in\Sigma^{<N}$ we have

${\left.\mkern-1.0mu\sigma^{w}\left(\varphi\left(t^{(\epsilon)}\right)\right)\right|}_{\Sigma^{<m}}={\left.\mkern-1.0mu\varphi\left(\tau^{w}\left(t^{(\epsilon)}\right)\right)\right|}_{\Sigma^{<m}}={\left.\mkern-1.0mu\tau^{w}(t^{(\epsilon)})\right|}_{\Sigma^{<m}}={\left.\mkern-1.0mus^{(w)}\right|}_{\Sigma^{<m}}={\left.\mkern-1.0mut^{(w)}\right|}_{\Sigma^{<m}}.$

The penultimate equality is by the definition of $\tau$ mappings and induction
as

$\tau^{w}\left(t^{(\epsilon)}\right)=\tau^{w}\left(s^{(\epsilon)}\right)=\tau^{w_{0}w_{1}\ldots}\left(s^{(\epsilon)}\right)=\tau^{w_{1}w_{2}\ldots}\left(s^{(w_{0})}\right)=\ldots=s^{(w)}.$

Therefore, the finite $[n]$-pseudo-orbit
$\left\\{t^{(w)}\right\\}_{w\in\Sigma^{<N}}$ is $[m]$-traced by some tree from
$X$, so $X$ has the finite pseudo-orbit-tracing property. ∎

We note two immediate corollaries.

######  Corollary 4.11.

If a tree-shift is topologically stable and has no isolated points then it is
of finite type.

######  Corollary 4.12.

A tree-shift with no isolated points is topologically stable if and only if it
is of finite type.

##  5\. Open tree-shifts

In [14] Parry proves that shift spaces such that the mapping
${\left.\mkern-1.0mu\sigma\right|}_{X}$ is open, are exactly shifts of finite
type. It appears that this result cannot be fully generalised to the tree-
shift case. In this section, we present an example of a tree-shift for which
all shift maps are open, but which is not of finite type. However we prove
that the converse implication holds: if a tree-shift $X$ is of finite type
then for every $i\in\Sigma$ the map
${\left.\mkern-1.0mu\sigma^{i}\right|}_{X}$ is open.

######  Theorem 5.1.

If a tree-shift $X$ is of finite type, then for every $i\in\Sigma$, the shift
map ${\left.\mkern-1.0mu\sigma^{i}\right|}_{X}$ is open.

###### Proof.

We generalise the proof presented in [14]. Let $X$ be a tree-shift of finite
type. We can assume that its forbidden set $\mathcal{F}$ is contained in
$\mathcal{B}_{n}(\mathcal{T})$ for some $n>0$. It suffices to show that for
any $b\in\mathcal{B}_{m}(X)$, where $m>n$, and $i\in\Sigma$, the set
$\sigma^{i}\left([b]\right)$ is open, because sets of this form constitute a
base for the topology of $X$. Let $t\in\sigma^{i}\left([b]\right)$ and let
$r\in[b]$ be such that $\sigma^{i}(r)=t$. We will prove that

$\left[{\left.\mkern-1.0mut\right|}_{\Sigma^{<m}}\right]\subseteq\sigma^{i}\left([b]\right).$

Let $s\in\left[{\left.\mkern-1.0mut\right|}_{\Sigma^{<m}}\right]$ and define
$q\colon\Sigma^{*}\to\mathcal{A}$ as follows. For $w\in\Sigma^{*}$, put
$q_{iw}=s_{w}$ and for words $u\in\Sigma^{*}$, that do not start with $i$, put
$q_{w}=r_{w}.$ It is then obvious that $\sigma^{i}(q)=s$. The tree $q$ is an
element of cylinder $[b]$ because

$q_{\epsilon}=r_{\epsilon}=b_{\epsilon},$
${\left.\mkern-1.0muq\right|}_{i\Sigma^{<m-1}}={\left.\mkern-1.0mus\right|}_{\Sigma^{<m-1}}={\left.\mkern-1.0mut\right|}_{\Sigma^{<m-1}}={\left.\mkern-1.0mu\sigma^{i}(r)\right|}_{\Sigma^{<m-1}}={\left.\mkern-1.0mur\right|}_{i\Sigma^{<m-1}}={\left.\mkern-1.0mub\right|}_{i\Sigma^{<m-1}},$

and for $j\in\Sigma\setminus\\{i\\}$ there is

${\left.\mkern-1.0muq\right|}_{j\Sigma^{<m-1}}={\left.\mkern-1.0mur\right|}_{j\Sigma^{<m-1}}={\left.\mkern-1.0mub\right|}_{j\Sigma^{<m-1}}.$

Therefore

${\left.\mkern-1.0muq\right|}_{\Sigma^{<m}}={\left.\mkern-1.0mub\right|}_{\Sigma^{<m}}.$

Also, for any $w\in\Sigma^{*}$, we have
${\left.\mkern-1.0muq\right|}_{w\Sigma^{<n}}\in\mathcal{B}_{n}(X)$. If $w$
starts with $i$ then $w=iu$ for some word $u$ and

${\left.\mkern-1.0muq\right|}_{iu\Sigma^{<n}}={\left.\mkern-1.0mus\right|}_{u\Sigma^{<n}}\in\mathcal{B}_{n}(X).$

If $w$ starts with some other letter then

${\left.\mkern-1.0muq\right|}_{w\Sigma^{<n}}={\left.\mkern-1.0mur\right|}_{w\Sigma^{<n}}\in\mathcal{B}_{n}(X).$

Finally, if $w=\epsilon$ then

${\left.\mkern-1.0muq\right|}_{\Sigma^{<n}}={\left.\mkern-1.0mub\right|}_{\Sigma^{<n}}\in\mathcal{B}_{n}(X).$

Thus $q\in X$ and so we get that $s\in\sigma^{i}([b])$.∎

Now, we will construct a tree-shift $X$ that is not of finite type, but for
any open set $U\subseteq X$ and $i\in\Sigma$, the set $\sigma^{i}(U)$ is open
in $X$.

######  Example 5.2.

We define the set $\mathcal{F}$ as follows. For $n>0$ and
$b\colon\\{0,1\\}^{<n}\to\\{0,1\\}$, let $b\in\mathcal{F}$ if and only if,
there are two different words $u,w\in\\{0,1\\}^{<n}$ of the same length, such
that $b_{w}=b_{u}=0$. We put $X=\mathcal{T}_{\mathcal{F}}$. We defined $X$ by
specifying its set of forbidden patterns, so $X$ is a tree-shift. A tree $t$
is in $X$ if and only if there are no two zeros in the same row of $t$.
Clearly $X$ is a non-empty tree-shift on $\Sigma=\\{0,1\\}$ over
$\mathcal{A}=\\{0,1\\}$.
Assume that $X$ is of finite type and its forbidden set $\mathcal{F}$ consists
of blocks of height $n>0$. Then we can define
$t\colon\\{0,1\\}^{*}\to\\{0,1\\}$ as follows. We put $t_{0^{n+1}}=0$,
$t_{1^{n+1}}=0$ and for words
$w\in\Sigma^{*}\setminus\left\\{0^{n+1},1^{n+1}\right\\}$, we set $t_{w}=1$.
Then, for any $w\in\Sigma^{*}$, at most one node labelled with 0 appears in
${\left.\mkern-1.0mut\right|}_{w\Sigma^{<n}}$, so it is in
$\mathcal{B}_{n}(X)$. This would mean that $t\in X$, but there are two words
of the same length, such that there are zeros at their nodes, so $t$ can not
be in $X$. This is a contradiction. Therefore $X$ is not of finite type.
We prove now that for every $i\in\Sigma$, the map
${\left.\mkern-1.0mu\sigma^{i}\right|}_{X}$ is open. Fix $i\in\Sigma$. We need
to show that for $U=[b]\subseteq X$, where $b\in\mathcal{B}(X)$, the set
$\sigma^{i}([b])$ is open in $X$. Let $b\in\mathcal{B}_{n}(X)$ for $n>0$ and
$t\in\sigma^{i}([b])$. Note that $t\in\sigma^{i}\left([b]\right)$ if and only
if $t={\left.\mkern-1.0mu\tilde{t}\right|}_{i\Sigma^{*}}$ for some
$\tilde{t}\in[b]$. We will prove that

$\left[{\left.\mkern-1.0mut\right|}_{\Sigma^{<n}}\right]\subseteq\sigma^{i}([b]).$

Let $s\in\left[{\left.\mkern-1.0mut\right|}_{\Sigma^{<n}}\right]$. We will
define $r\colon\\{0,1\\}^{*}\to\\{0,1\\}$ as follows. For $w\in\Sigma^{*}$,
let $r_{iw}=s_{w}$. For $w\in\Sigma^{<n}$ that does not start with $i$, let
$r_{w}=b_{w}$ and for all words $w\in\Sigma^{*}\setminus\Sigma^{<n}$ that do
not start with $i$, let $r_{w}=1$. The tree $r$ constructed in this way is an
element of $[b]$. The first observation is that
${\left.\mkern-1.0mur\right|}_{\Sigma^{<n}}=b$ as if $w\in\Sigma^{<n}$ does
not start with $i$ then by definition $r_{w}=b_{w}$ and if $w=i\tilde{w}$ for
some $\tilde{w}\in\Sigma^{<n-1}$ then

$r_{w}=s_{\tilde{w}}=t_{\tilde{w}}=\tilde{t}_{i\tilde{w}}=\tilde{t}_{w}=b_{w}.$

Also, $r\in X$ as it can be proved by considering a few cases. If $m\leq n$
then $r_{u}=b_{u}$ and $r_{v}=b_{v}$ so they can not be both 0, because
$b\in\mathcal{B}(X)$. If $m>n$ and words $u,v$ both start with $i$ then they
can be expressed as $u=i\tilde{u},v=i\tilde{v}$ for some
$\tilde{u},\tilde{v}\in\Sigma^{m-1}$. Then by the definition of $r$ we have
$r_{u}=s_{\tilde{u}}$ and $r_{v}=s_{\tilde{v}}$. As $s\in X$, the labels of
$s$ at $\tilde{u}$ and $\tilde{v}$ can not be both 0 and we are done with this
case. If $m>n$ and one of words $u,v$ does not start with $i$ then this word
is 1. Therefore $r$ is an element of $[b]$ and of course we have
$s=\sigma^{i}(r)$ so $s\in\sigma^{i}([b])$.

##  6\. Conclusion

We proved that for a tree-shift being of finite type is equivalent to the
pseudo-orbit-tracing property and implies topological stability. To obtain the
pseudo-orbit-tracing property from topological stability, we assumed that the
considered tree-shift is perfect (has no isolated points). To the best of the
author’s knowledge, it is unknown if the equivalence between being of finite
type and topological stability also holds without assuming that the shift
space is perfect. Additionally, we proved that if $X$ is a tree-shift of
finite type then the restricted shift maps
$\left\\{{\left.\mkern-1.0mu\sigma^{i}\right|}_{X}\right\\}_{i\in\Sigma}$ are
open, but the converse fails when we consider tree-shifts with alphabet
$\Sigma$ consisting of more than one symbol. Therefore the relations between
the notions can be summarized by the following diagram (here TS stands for
”topologically stable, POTP for ”pseudo-orbit-tracing property” and FT for ”of
finite type”)

${X\text{ is open}}$${X\text{ is FT}}$${X\text{ has POTP}}$${X\text{ is
TS}}$${X\text{ is perfect and TS}}$

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