# Equilibrium States for Random Zooming Systems

Rafael A. Bilbao  Rafael A. Bilbao, Universidad Pedagógica y Tecnológica de
Colombia, Avenida Central del Norte 39-115, Sede Central Tunja, Boyacá,
150003, Colombia<EMAIL_ADDRESS>,  Marlon Oliveira  Marlon
Oliveira, Departamento de Matemática e Informática, Universidade Estadual do
Maranhão, 65055-310 São Luís, Brazil<EMAIL_ADDRESS>and
Eduardo Santana  Eduardo Santana, Universidade Federal de Alagoas, 57200-000
Penedo, Brazil<EMAIL_ADDRESS>
(Date: August 28, 2024)

###### Abstract.

In this work, we introduce the notion of random zooming systems and prove the
existence of equilibrium states for which we call random zooming potentials,
that include the hyperbolic ones, possibly with the presence of a critical
set. The proof follows ideas of the work [4]. With a mild condition, we obtain
uniqueness. We also prove that the classes of random zooming potentials and
random hyperbolic potentials are equivalent and also contain the null
potential, giving measures of maximal entropy.

The third author was partially supported by CNPq, under the project with
reference 409198/2021-8.

##  1\. Introduction

The theory of equilibrium states on dynamical systems was firstly developed by
Sinai, Ruelle and Bowen in the sixties and seventies. It was based on
applications of techniques of Statistical Mechanics to smooth dynamics. Given
a continuous map $f:M\to M$ on a compact metric space $M$ and a continuous
potential $\phi:M\to\mathbb{R}$, an equilibrium state is an invariant measure
that satisfies a variational principle, that is, a measure $\mu$ such that

$\displaystyle h_{\mu}(f)+\int\phi
d\mu=\sup_{\eta\in\mathcal{M}_{f}(M)}\bigg{\\{}h_{\eta}(f)+\int\phi
d\eta\bigg{\\}},$

where $\mathcal{M}_{f}(M)$ is the set of $f$-invariant probabilities on $M$
and $h_{\eta}(f)$ is the so-called metric entropy of $\eta$.

In the context of uniform hyperbolicity, which includes uniformly expanding
maps, equilibrium states do exist and are unique if the potential is Hölder
continuous and the map is transitive. In addition, the theory for finite
shifts was developed and used to achieve the results for smooth dynamics.

Beyond uniform hyperbolicity, the theory is still far from complete. It was
studied by several authors, including Bruin, Keller, Demers, Li, Rivera-
Letelier, Iommi and Todd [6, 9, 10, 13] for interval maps; Denker and Urbanski
[8] for rational maps; Leplaideur, Oliveira and Rios [14] for partially
hyperbolic horseshoes; Buzzi, Sarig and Yuri [7, 25], for countable Markov
shifts and for piecewise expanding maps in one and higher dimensions. For
local diffeomorphisms with some kind of non-uniform expansion, there are
results due to Oliveira [15]; Arbieto, Matheus and Oliveira [4]; Varandas and
Viana [23]. All of whom proved the existence and uniqueness of equilibrium
states for potentials with low oscillation. Also, for this type of maps, Ramos
and Viana [20] proved it for potentials so-called hyperbolic, which includes
the previous ones. The hyperbolicity of the potential is characterized by the
fact that the pressure emanates from the hyperbolic region. In all these
studies the maps do not have the presence of critical sets and recently,
Alves, Oliveira and Santana proved the existence of at most finitely many
equilibrium states for hyperbolic potentials, possible with the presence of a
critical set (see [3]). More recently, Santana completed this by showing
uniqueness in [21]. Moreover, in this work it is proved that the class of
hyperbolic potentials is equivalent to the class of continuous zooming
potential (which satisfies a key inequality between free energies). It
includes the null potential, which implies existence and uniqueness of measure
of maximal entropy. A similar work is developed in [19], where they deal with
expanding measures and potentials.

In the context of random dynamics, some works has established existence and
others uniqueness. We can cite [4, 5, 11, 22]. All these works deal with
dynamics with absence of critical sets. In this work, we generalize some of
these results and extend them to the context of zooming contractions,
obtaining random zooming systems. Based on the definition of zooming maps,
introduced in [18] and developed equilibrium states for the deterministic
context in [21], we introduce random zooming systems.

We define what we mean by random zooming maps (which generalizes both
deterministic zooming systems and random non-uniformly expanding maps), random
zooming potentials and follow ideas of [4] to obtain equilibrium states. With
a mild condition, we obtain uniqueness. Moreover, based on ideas of [21] we
prove that the class of random zooming potentials is equivalent to the class
of random hyperbolic potentials and contain the null potential, giving
measures of maximal entropy.

In the Example section, we give the class of random Viana maps and construct
random zooming potentials for maps with the presence of a fixed point in the
random zooming set, which is the case of the random Viana maps. Finally, we
prove that for random systems which have equilibrium states that are not
measures of maximal entropy at the same time, we have the null potential as a
random zooming potential. We also present an example of random zooming system
which is not random non-uniformly expanding.

##  2\. Preliminaries and Main Result

###  2.1. Random Maps and Invariant Measures

Let $M$ be a connected compact metric space and
$\mathcal{C}^{0}:=\mathcal{C}^{0}(M,M)$ the space of continuous maps on $M$.
Let $(\Omega,T,\mathbb{P})$ be a measure preserving system, where
$T:\Omega\to\Omega$ is $\mathbb{P}$-invariant (where $\mathbb{P}$ is a Borel
measure) and $\Omega$ is a Polish space, i.e., $\Omega$ is a complete
separable metric space. By a random transformation we mean a continuos map
$f:\Omega\to\mathcal{C}^{0}$. Then, we define the skew-product generated by
$f$:

$\begin{array}[]{cccc}F\ :&\\!\Omega\times
M&\\!\longrightarrow&\\!\Omega\times M\\\
&\\!(w,x)&\\!\longmapsto&\\!(T(w),f_{w}(x))\end{array}$

where the iterates of $F$ are given by

$F^{n}(w,x)=(T^{n}(w),f^{n}_{w}(x))$

with,

$f^{n}_{w}(x):=f_{T^{n-1}(w)}\circ\dots\circ f_{w}(x).$

We denote by $\mathcal{P}_{\mathbb{P}}(\Omega\times M)$ the space of
probability measures $\mu$ on $\Omega\times M$ such that the marginal of $\mu$
on $\Omega$ is $\mathbb{P}$. Let $\mathcal{M}(F,\Omega\times
M)\subset\mathcal{P}_{\mathbb{P}}(\Omega\times M)$ the space of probability
measure $\mu$ that are $F$-invariant. We denote the fiber at $w\in\Omega$ by
$M_{w}=\\{w\\}\times M$.

Because $M$ is compact, invariant measures always exist and the property of
$\mathbb{P}$ being the marginal on $\Omega$ of an invariant measure can be
characterized by its disintegration:

$d\mu(w,x)=d\mu_{w}(x)d\mathbb{P}(x).$

$\mu_{w}$ are called sample measures of $\mu$.

###  2.2. Random Zooming Systems

Based on the definition of zooming maps, introduced in [18] and developed
equilibrium states for the deterministic context in [21], we introduce random
zooming systems.

######  Definition 1.

(Zooming contractions). A zooming contraction is a sequence of functions
$\alpha_{n}:[0,+\infty)\to[0,+\infty)$ such that

  * •

$\alpha_{n}(r)<r,\text{for all}\,\,n\in\mathbb{N},\text{for all}\,\,r>0.$

  * •

$\alpha_{n}(r)<\alpha_{n}(s),\,\,if\,\,0<r<s,\text{for
all}\,\,n\in\mathbb{N}$.

  * •

$\alpha_{m}\circ\alpha_{n}(r)\leq\alpha_{m+n}(r),\text{for
all}\,\,r>0,\text{for all}\,\,m,n\in\mathbb{N}$.

  * •

$\displaystyle\sup_{r\in(0,1)}\sum_{n=1}^{\infty}\alpha_{n}(r)<\infty$.

As defined in [19], we call the contraction $(\alpha_{n})_{n}$ exponential if
$\alpha_{n}(r)=e^{-\lambda n}r$ for some $\lambda>0$ and Lipschitz if
$\alpha_{n}(r)=a_{n}r$ with $0\leq a_{n}<1,a_{m}a_{n}\leq a_{m+n}$ and
$\sum_{n=1}^{\infty}a_{n}<\infty$. In particular, every exponential
contraction is Lipschitz. We can also have the example with
$a_{n}=(n+b)^{-a},a>1,b>0$.

We define a random $(\alpha,\delta)$-zooming time for $(w,x)\in\Omega\times
M$. Fix a zooming contraction $\alpha$, $\delta>0$ and $(w,x)\in\Omega\times
M$. We say that $n\in\mathbb{N}$ is a $(\alpha,\delta)$-random zooming time
for $(w,x)$ if

  * •

There exists a neighbourhood $V_{n}(w,x)\subset\\{w\\}\times M$ of $(w,x)$
such that $f^{n}_{w}$ sends $V_{n}(w,x)$ homeomorphically onto the ball
$B_{\delta}(f^{n}_{w}(x))\subset\\{T^{n}(w)\\}\times M$;

  * •

$d(f^{i}_{w}(y),f^{i}_{w}(z))\leq\alpha_{n-i}(d(f^{n}_{w}(y),f^{n}_{w}(z))),\forall\,\,y,z\in
V_{n}(w,x),\forall\,\,0\leq i\leq n-1.$

We require that it holds for $\mathbb{P}$ almost every $w\in\Omega$.

The sets $V_{n}(w,x)$ are called random zooming pre-balls and their images
$f^{n}_{w}(V_{n}(w,x))=B_{\delta}(f^{n}_{w}(x))$, random zooming balls.

We observe that if $n$ is a $(\alpha,\delta)$-random zooming time for $(w,x)$,
then $n$ is a $(\alpha,\delta^{\prime})$-random zooming time for $(w,x)$, for
every $0<\delta^{\prime}<\delta$.

We say that $(w,x)\in\Omega\times M$ has positive frequency of random zooming
times if

(1) $\limsup_{n\to\infty}\frac{1}{n}\\#\bigg{\\{}0\leq j\leq
n-1\,\,|\,\,j\,\,\text{is a random zooming time for}\,\,(w,x)\bigg{\\}}>0,$

and define the random zooming set

$\Lambda=\\{(w,x)\in\Omega\times M\mid\text{the frequency
of}\,\,(\alpha,\delta)-\text{random zooming times of}\,\,(w,x)\,\,\text{is
positive}\\}.$

We say that a Borel probability measure $\mu$ on $\Omega\times M$ is random
zooming if $\mu(\Lambda)=1$.

A map with an associated random zooming measure is called a _random zooming
map_. We will always consider random zooming maps admitting invariant random
zooming measures.

###  2.3. Entropy and Topological Pressure

We start by the definition of entropy and topological pressure for random
transformations. The reader can consult more results and properties in Kifer
[11] and Liu [12].

Let $\mu\in\mathcal{M}(F,\Omega\times M)$ be an $F$-invariant measure. Given a
finite measurable partition $\xi$ of $M$ we set

$h_{\mu}(F|T;\xi):=\lim_{n\rightarrow+\infty}\frac{1}{n}\int_{\Omega}H_{\mu_{w}}\left(\bigvee_{j=0}^{n-1}f_{w}^{-j}(\xi)\right)d\mathbb{P}(w)$

where $H_{\nu}(\xi)=-\sum_{P\in\xi}\nu(P)\log\nu(P)$ for a finite partition
$\xi$ and $\mu_{w}$ is the sample measure of $\mu.$ The _entropy_ of
$(F|T,\mu)$ is

$h_{\mu}(F|T):=\sup_{\xi}\\{h_{\mu}(F|T;\xi)\\}$

where the supremum is taken over all finite measurable partitions of $M$.

Denote by $\mathbb{L}^{1}_{\mathbb{P}}(\Omega,C^{0}(M))$ the space of all
measurable functions $\phi:\Omega\times M\rightarrow\mathbb{R}$ such that
$\phi_{w}:M\rightarrow\mathbb{R}$ defined by $\phi_{w}(x):=\phi(w,x)$ is
continuous for all $w\in\Omega$ and
$\lVert\phi\rVert_{1}=\int_{\Omega}\lVert\phi_{w}\rVert_{\infty}\,d\mathbb{P}(w)<+\infty$.

Fix $w\in\Omega$. Given $\varepsilon>0$ and an integer $n\geq 1$, we say that
a subset $F_{n}\subseteq M$ is $(w,n,\varepsilon)-$separated if for every two
distinct points $y,z\in F_{n}$ there exists some $j\in\\{0,1,...,n-1\\}$ such
that $d(f^{j}_{w}(y),f^{j}_{w}(z))>\varepsilon$.

For $\phi\in\mathbb{L}^{1}_{\mathbb{P}}(\Omega,C^{0}(M))$, $\varepsilon>0$ and
$n\geq 1$ we consider

$P_{F|T}(\phi)(w,n,\varepsilon)=\sup\left\\{\sum_{y\in
F_{n}}e^{S_{n}\phi(w,y)}\ ;\ F_{n}\ \mbox{is a}\
(w,n,\varepsilon)-\mbox{separated set}\right\\}$

where $S_{n}\phi(w,y):=\sum^{n-1}_{j=0}\phi_{\theta^{j}(w)}(f^{j}_{w}(y))$.

The _random topological pressure_ of $\phi$ relative to $\theta$ is defined by

$P_{F|T}(\phi)=\lim_{\varepsilon\rightarrow
0}\limsup_{n\rightarrow\infty}\frac{1}{n}\int_{\Omega}\log
P_{F|T}(\phi)(w,n,\varepsilon)\,d\mathbb{P}(w).$

Thus it is well defined the pressure map as follows

$\begin{array}[]{cccc}P_{F|T}\
:&\\!\mathbb{L}^{1}_{\mathbb{P}}(\Omega,C^{0}(M))&\\!\longrightarrow&\\!\mathbb{R}\cup\\{\infty\\}\\\
&\\!\phi&\\!\longmapsto&\\!P_{F|T}(\phi)\end{array}$

In particular, the topological entropy of $F$ relative to $T$ is
$h_{top}(F|T)=P_{F|T}(0)$.

The topological pressure and the entropy are related by the well known
Variational Principle. The reader can see a proof of this result in [12].

######  Theorem 1 (Variational Principle).

Assume that $(\Omega,\mathbb{P})$ is a Lebesgue space. Then for any
$\phi\in\mathbb{L}^{1}_{\mathbb{P}}(\Omega,C^{0}(M))$ we have

(2)
$P_{F|T}(\phi)=\sup_{\mu\in\mathcal{M}_{\mathbb{P}}(F)}\biggl{(}h_{\mu}(F|T)+\int\phi\
d\mu\biggr{)}.$

Moreover, when $\mathbb{P}$ is ergodic, we can consider the supremum over
ergodic measures.

Motivated by the Variational Principle, we say that an $F$-invariant measure
$\mu\in\mathcal{M}(\Omega\times M)$ is an _equilibrium state_ for $(F|T,\phi)$
relative to $T$ if the supremum (2) is attained by $\mu$, i.e.,

$P_{F|T}(\phi)=h_{\mu}(F|T)+\int\phi\ d\mu.$

###  2.4. Topological Pressure and Variational Principle to noncompact set

Let $w\in\Omega$ and consider $\mathcal{U}$ a finite open cover of $M$. Denote
by $\mathcal{S}_{n}(\mathcal{U})$ the set of all strings
${\bf{U}}=\\{U_{i_{0}},\cdots,U_{i_{n-1}}\,;\,U_{i_{j}}\in\mathcal{U}\\}$ of
length $n=n({\bf{U}})$ and put
$\mathcal{S}=\mathcal{S}(\mathcal{U})=\bigcup_{n\geq
0}\mathcal{S}_{n}(\mathcal{U})$. Given a string
${\bf{U}}=\\{U_{i_{0}},\cdots,U_{i_{n-1}}\\}\in\mathcal{S}(\mathcal{U})$, we
consider the cylinder

$X_{w}=X_{w}({\bf{U}}):=\\{x\in M\,\,;\,\,f^{j}_{w}(x)\in
U_{i_{j}}\,\,\mbox{for}\,\,j=0,\cdots,n({\bf U})-1\\}.$

Let $\mathcal{F}_{(N,w)}$ be the collection of all cylinders of depth at least
$N$, i.e.,

$\mathcal{F}_{(N,w)}=\mathcal{F}_{(N,w)}(\mathcal{U})=\\{X_{w}({\bf{U}})\,\,;\,\,{\bf{U}}\in\mathcal{S}_{n}(\mathcal{U})\,\,\mbox{for}\,\,n\geq
N\\}.$

For $\beta\in\mathbb{R}$ and
$\phi\in\mathbb{L}^{1}_{\mathbb{P}}(\Omega,C^{0}(M))$ let

(3)
$m_{\beta}(w,\phi,F|T,\mathcal{U},N)=\displaystyle{\inf_{\mathcal{F}}}\left\\{\sum_{X_{w}\in\mathcal{F}_{(N,w)}}e^{-\beta
n({\bf{U}})+S_{n({\bf{U}})}\phi(X_{w})}\right\\},$

where $S_{n({\bf{U}})}\phi(X_{w})=\sup_{y\in X_{w}}\sum_{j=0}^{n({\bf
U})-1}\phi_{T^{j}(w)}(f^{j}_{w}(y))$ and the infimum is taken over all finite
families $\mathcal{F}$ of $\mathcal{F}_{(N,w)}$ in order that (3) is
measurable in $w$ (see e.g. Section 9 of [22]). As $N$ goes to infinity we
define

$m_{\beta}(w,\phi,F|T,\mathcal{U})=\lim_{N\to\infty}m_{\beta}(w,\phi,F|T,\mathcal{U},N).$

The existence of the limit above is guaranteed by the function
$m_{\beta}(w,\phi,F|T,\mathcal{U},N)$ to be increasing with $N$. Taking the
infimum over $\beta$ we call

$P_{F|T}(w,\phi,\mathcal{U})=\inf\\{\beta:m_{\beta}(w,\phi,F|T,\mathcal{U})=0\\}.$

Let $|\mathcal{U}|=\max\\{\mbox{diam}U_{i}\,\,;\,\,U_{i}\subset\mathcal{U}\\}$
be the diameter of the cover $\mathcal{U}$ and consider

$P_{F|T}(w,\phi)=\lim_{|\mathcal{U}|\rightarrow
0}{P_{F|T}(w,\phi,\mathcal{U})}.$

In [Theorem 11.1, [17]] it was showed that this quantity is well defined and
does not depend on the cover $\mathcal{U}$. Moreover, since all quantities
defined above are measurable functions of $w\in\Omega$ (see e.g. Section 9 of
[22]), we can define the _random topological pressure_ of $(F|T,\phi)$ as

$P_{F|T}(\phi)=\int P_{F|T}(w,\phi)\ d\mathbb{P}(w).$

In the following we present another way to define the random topological
pressure. We fix $w\in\Omega$ and $\varepsilon>0$. For $n\in\mathbb{N}$, $x\in
M$, let $B_{w}(x,n,\varepsilon)$ be the dynamic ball

$B_{w}(x,n,\varepsilon):=\\{y\in M:d(f_{w}^{j}(x),f_{w}^{j}(y))<\varepsilon,\
\mbox{ for}\ \ 0\leq j\leq n\\}.$

We denote by $G_{(N,w)}$ the collection of dynamic balls:

$G_{(N,w)}:=\\{B_{w}(x,n,\varepsilon):x\in M\ \mbox{and}\ n\geq N\\}.$

Let $U_{w}$ be a finite or countable family of $G_{(N,w)}$ which covers $M$.
For every $\beta\in\mathbb{R}$ and
$\phi\in\mathbb{L}^{1}_{\mathbb{P}}(\Omega,C^{0}(M))$ let

$m_{\beta}(w,\phi,F|T,\varepsilon,N)=\inf_{U_{w}\subset
G_{(N,w)}}\left\\{\sum_{B_{w}(x,n,\varepsilon)\in U_{w}}e^{-\beta
n+S_{n}\phi(B_{w}(x,n,\varepsilon))}\right\\},$

where $S_{n}\phi(B_{w}(x,n,\varepsilon))=\sup_{y\in
B_{w}(x,n,\varepsilon)}\sum_{j=0}^{n-1}\phi_{T^{j}(w)}(f^{j}_{w}(y))$. When
$N$ goes to infinity we consider

$m_{\beta}(w,\phi,F|T,\varepsilon)=\lim_{N\to\infty}m_{\beta}(w,\phi,F|T,\varepsilon,N).$

and taking the infimum over $\beta$ we define

$P_{F|T}(w,\phi,\varepsilon)=\inf\\{\beta:m_{\beta}(w,\phi,F|T,\varepsilon)=0\\}.$

Since $P_{F|T}(w,\phi,\varepsilon)$ is decreasing on $\varepsilon$ we can take
the limit

$P_{F|T}(w,\phi)=\lim_{\varepsilon\rightarrow
0}{P_{F|T}(w,\phi,\varepsilon)}.$

Now if we consider a finite open cover $\mathcal{U}$ of $M$ with Lebesgue
number $\varepsilon(\mathcal{U})$ we have

$B_{w}(x,n({\bf U}),\frac{1}{2}\varepsilon)\subset X_{w}({\bf U})\subset
B_{w}(x,n({\bf U}),2|\mathcal{U}|)$

which implies that

$P_{F|T}(w,\phi)=\lim_{\varepsilon\rightarrow
0}{P_{F|T}(w,\phi,\varepsilon)}=\lim_{|\mathcal{U}|\rightarrow
0}{P_{F|T}(w,\phi,\mathcal{U})}.$

Therefore the definition of random topological pressure via coverings and via
dynamic balls coincides.

The topological pressure and the entropy are related by the well known
Variational Principle. The reader can see a proof of this result in [22].

######  Theorem 2.

(Variational Principle to noncompact sets). Let
$\phi\in\mathbb{L}^{1}_{\mathbb{P}}(\Omega,C^{0}(M))$ and let
$\mu\in\mathcal{M}(\Omega\times M)$ be a probability measure and
$\Lambda\subset\Omega\times M$ such that $F(\Lambda)\subseteq\Lambda$. Then

$P_{F|T}(\phi,\Lambda)\geq\sup_{\mu(\Lambda)=1}\left\\{h_{\mu}(F|T)+\int_{\Lambda}\phi
d\mu\right\\}.$

###  2.5. Random Zooming Potentials

Analogouly to what is given in [19], we introduce the following definitions in
the random context.

Let $\mathcal{Z}$ be the set of random zooming measures on $\Omega\times M$.
We say that a function $\phi\in\mathbb{L}^{1}_{\mathbb{P}}(\Omega,C^{0}(M))$
is a _random zooming potential_ if the following inequality holds:

(4) $\displaystyle\sup_{\nu\in\mathcal{Z}^{c}}\bigg{\\{}h_{\nu}(F|T)+\int\phi
d\nu\bigg{\\}}<\sup_{\mu\in\mathcal{Z}}\bigg{\\{}h_{\mu}(F|T)+\int\phi
d\mu\bigg{\\}}.$

If $P(\phi)$ denotes the topological pressure of $\phi$, by the Variational
Principle, we have:

$P(\phi)=\sup_{\mu\in\mathcal{M}(\Omega\times
M)}\bigg{\\{}h_{\mu}(F|T)+\int\phi
d\mu\bigg{\\}}=\sup_{\mu\in\mathcal{Z}}\bigg{\\{}h_{\mu}(F|T)+\int\phi
d\mu\bigg{\\}}.$

It means that, if the there exists some equilibrium state for a random zooming
potential, the equilibrium states must be random zooming measures.

###  2.6. Random Hyperbolic Potentials

We say that a function $\phi\in\mathbb{L}^{1}_{\mathbb{P}}(\Omega,C^{0}(M))$
is a _random hyperbolic potential_ if the topological pressure $P_{F|T}(\phi)$
is concentrated at the random zooming set $\Lambda$, that is, if the following
inequality holds:

(5) $P_{F|T}(\phi,\Lambda^{c})<P_{F|T}(\phi,\Lambda)=P_{F|T}(\phi).$

As a consequence of the Theorem 2 (Variational Principle), we have the
following inequality

$\displaystyle\sup_{\nu\in\mathcal{Z}^{c}}\bigg{\\{}h_{\nu}(F|T)+\int\phi
d\nu\bigg{\\}}\leq
P_{F|T}(\phi,\Lambda^{c})<P_{F|T}(\phi,\Lambda)=P_{F|T}(\phi)=\sup_{\mu\in\mathcal{Z}}\bigg{\\{}h_{\mu}(F|T)+\int\phi
d\mu\bigg{\\}}.$

It means that every random hyperbolic potential is a random zooming potential.

###  2.7. Main Result

Our main result concerns the existence and uniqueness of equilibrium states
for random zooming systems.

######  Theorem A.

Let $F:\Omega\times M\to\Omega\times M$ be a random zooming map and
$\phi:\Omega\times M\to\mathbb{R}$ a random zooming potential. Then,

  * •

There exists some equilibrium state and every equilibrium state is a random
zooming measure.

  * •

Suppose that the marginal $\mathbb{P}$ is ergodic with full support. Also
suppose that the map $F:\Omega\times M\to\Omega\times M$ is such that the set
$\\{f^{n}_{w}(x)\,\,|\,\,n\in\mathbb{N}\\}$ is dense on $M$, for every
$(w,x)\in\Lambda\cap M_{w}$, $\mathbb{P}$-a.e. $w\in\Omega$. In this case, we
have uniqueness of equilibrium state.

As a corollary of our main result concerns of the equivalence between the
classes of random zooming potentials and random hyperbolic potentials.

######  Corollary 1.

We have that every random zooming potential is also a random hyperbolic
potential. These two classes of potentials are then equivalent.

##  3\. Proof of Theorem A

The following results can be found in [4]. The first of them shows the
expansivity of the dynamics on the fibers. In the Lemma 2 the existence of a
generating partition is given over the set of measures $\mu$ that are in
$\mathcal{Z}$, which is the set of random zooming measures on $\Omega\times
M$. Consequently, the entropy over these measurements is equal; to the entropy
considering the generating partition. Finally, the upper semicontinuity is
obtained over a measure $\mu_{0}$.

######  Lemma 1.

Let $\epsilon>0$ and define

$A_{\epsilon}(w,x)=\\{y\mid d(f^{n}_{w}(x),f^{n}_{w}(y)\leq\epsilon,\forall
n\geq 0\\}.$

For every $\epsilon\leq\delta$ we have

$A_{\epsilon}(w,x)=\\{x\\},$

$\mathbb{P}$-almost every $w\in\Omega$.

###### Proof.

By equation 1 we have that every point $(w,x)\in\Lambda$ possesses infinitely
many $(\alpha,\delta)$-zooming times $n_{i}=n_{i}(w,x)\in\mathbb{N}$. For each
$w\in\Omega$, set $\Lambda_{w}:=\\{x\mid(w,x)\in\Lambda\\}$. Then,
$\mathbb{P}$-almost every $w\in\Omega$ we have $\mu_{w}(\Lambda_{w})=1$ and an
infinity of $(\alpha,\delta)$-zooming times $\mu_{w}$-almost every
$x\in\Lambda_{w}$. Also, for every $\epsilon\leq\delta$ we have that

$d(x,z)\leq\alpha_{n_{i}}(d(f^{n_{i}}_{w}(x),f^{n_{i}}_{w}(z)))\leq\alpha_{n_{i}}(\epsilon),$

because the contractions $\alpha_{n}$ are increasing functions. Also, since
the following holds
$\displaystyle\sup_{r\in(0,1)}\sum_{n=1}^{\infty}\alpha_{n}(r)<\infty$, we
have that for $\epsilon\in(0,1)$ we obtain $\alpha_{n_{i}}(\epsilon)\to 0$. It
implies that $d(x,z)=0$ and we are done. ∎

Let $\mathcal{P}$ be a partition of $M$ in measurable sets with diameter less
than $\delta$. From the above lemma, we state the following lemmas, which are
useful for proving the main theorem and whose proofs are found in [4]:

######  Lemma 2.

[4, Lemma 6.7] Let $\mathcal{P}$ be a partition of $M$ in measurable sets with
diameter less that $\delta$. Then, $\mathcal{P}$ is a generating partition for
every $\mu\in\mathcal{Z}$.

######  Corollary 2.

[4, Lemma 6.8] For every $\mu\in\mathcal{Z}$ we have
$h_{\mu}(F|T)=h_{\mu}(F,\mathcal{P})$.

######  Lemma 3.

[4, Lemma 6.9] The map $\mu\to h_{\mu}(F,\mathcal{P})$ is upper semi-
continuous at $\mu_{0}$ measure such that $(\mu_{0})_{w}(\partial P)=0$ for
$\mathbb{P}$-almost everywhere $w\in\Omega$, $P\in\mathcal{P}$.

Now, we are able to prove the Theorem A.

###### Proof.

Since the potential $\phi:\Omega\times M\to\mathbb{R}$ is random zooming, we
can obtain a sequence of measures $\mu_{k}\in\mathcal{Z}$ such that

$h_{\mu_{k}}(F|T)+\int\phi d\mu_{k}\to P(\phi).$

Fix a partition $\mathcal{P}$ with diameter less than $\delta$ and for
$w$-a.e., $\mu_{w}(\partial P)=0$, for any $P\in\mathcal{P}$. By the above
Corollary 2, we have that $h_{\mu_{k}}(F|T)=h_{\mu_{k}}(F,\mathcal{P})$. Then,
we obtain

$P(\phi)=\limsup_{k\to\infty}\bigg{\\{}h_{\mu_{k}}(F|T)+\int\phi
d\mu_{k}\bigg{\\}}.$

We know that the map $\eta\to h_{\eta}(F,\mathcal{P})$ is upper semi-
continuous over $\mathcal{Z}$ (also over $\overline{\mathcal{Z}}$). By
compacity of the space of probabilities, we can suppose that there exists a
measure $\mu\in\overline{\mathcal{Z}}$ such that

$P(\phi)=\limsup_{k\to\infty}\bigg{\\{}h_{\mu_{k}}(F,\mathcal{P})+\int\phi
d\mu_{k}\bigg{\\}}\leq h_{\mu}(F,\mathcal{P})+\int\phi d\mu\leq P(\phi).$

It means that $\mu$ is an equilibrium state for $\phi$. ∎

With the assumptions of the following lemma, we obtain the uniqueness of
equilibrium state.

######  Lemma 4.

Suppose that the marginal $\mathbb{P}$ is ergodic with full support. Also
suppose that the map $F:\Omega\times M\to\Omega\times M$ is such that the set
$\\{f^{n}_{w}(x)\,\,|\,\,n\in\mathbb{N}\\}$ is dense on $M$, for every
$(w,x)\in\Lambda\cap M_{w}$, $\mathbb{P}$-a.e. $w\in\Omega$. Then, we have
uniqueness of equilibrium state.

###### Proof.

Let us suppose, by contradiction, that there exist two ergodic equilibrium
states $\mu\neq\nu$. Since $\mu(\Lambda)=1=\nu(\Lambda)$, because they are
random zooming measures, we have that $\mu_{w}(\Lambda\cap M_{w})=1$ and
$\nu_{w}(\Lambda\cap M_{w})=1$, $\mathbb{P}$-a.e. $w\in\Omega$. There exists
$w_{0}\in\Omega$ such that $\mu_{w_{0}}(\Lambda\cap
M_{w_{0}})=1=\nu_{w_{0}}(\Lambda\cap M_{w_{0}})$. We can also take $w_{0}$
such that the set $\\{f^{n}_{w_{0}}(x)\,\,|\,\,n\in\mathbb{N}\\}$ is dense on
$M$.

Fix $w_{0}\in\Omega$. Given $\epsilon>0$, Let $B_{0}\subset M$ be a ball with
radius $\epsilon$. Take $(w_{0},x_{0})\in\Lambda\cap M_{w_{0}}$. There exists
$n_{0}\in\mathbb{N}$ such that

$f^{n_{0}}_{w_{0}}(x_{0})\in B_{0}\subset M_{T^{n_{0}}(w_{0})}\implies
x_{0}\in f^{-n_{0}}_{T^{n_{0}}(w_{0})}(B_{0})\subset M_{w_{0}}.$

It means that

$\Lambda\cap
M_{w_{0}}\subset\bigcup_{i=1}^{\infty}f^{-i}_{T^{i}(w_{0})}(B_{0}).$

Since $\mu_{w_{0}}(\Lambda\cap M_{w_{0}})=1=\nu_{w_{0}}(\Lambda\cap
M_{w_{0}})$, there exist $r,s\in\mathbb{N}$ such that

$\mu_{w_{0}}(f^{-r}_{T^{r}(w_{0})}(B_{0}))>0,\nu_{w_{0}}(f^{-s}_{T^{s}(w_{0})}(B_{0}))>0,$

which implies, by invariance, that

$\mu_{T^{r}(w_{0})}(B_{0})>0,\nu_{T^{s}(w_{0})}(B_{0})>0\implies\mu_{T^{t}(w_{0})}(B_{0})>0,\nu_{T^{t}(w_{0})}(B_{0})>0,t\geq\max\\{r,s\\}.$

We claim that
$\mathbb{P}(\\{w\in\Omega\,\,|\,\,\mu_{w}(B_{0})>0,\nu_{w}(B_{0})>0\\})=1$. In
fact, this set is positively invariant. Since $\mathbb{P}$ is ergodic, we have
$\mathbb{P}(\\{w\in\Omega\,\,|\,\,\mu_{w}(B_{0})>0,\nu_{w}(B_{0})>0\\})=0,1$.
If $\mathbb{P}(\\{w\in\Omega\,\,|\,\,\mu_{w}(B_{0})>0,\nu_{w}(B_{0})>0\\})=0$,
we would have

$w_{0}\in\bigcup_{i=1}^{\infty}T^{-i}(\\{w\in\Omega\,\,|\,\,\mu_{w}(B_{0})>0,\mu_{w}(B_{0})>0\\})\,\,\mathbb{P}-\text{a.e.}\,\,w_{0}\in\Omega\,\,(*),$

and the union in $(*)$ with zero measure $\mathbb{P}$, by invariance. This is
not possible and we obtain
$\mathbb{P}(\\{w\in\Omega\,\,|\,\,\mu_{w}(B_{0})>0,\nu_{w}(B_{0})>0\\})=1$.
So, $\mu(A\times B_{0})>0,\,\,\nu(A\times B_{0})>0$ for every set
$A\subset\Omega$ such that $\mathbb{P}(A)>0$ and we obtain
$\text{supp}(\mu)=\Omega\times M=\text{supp}(\nu)$ because
$\text{supp}(\mathbb{P})=\Omega$. Since both measures are ergodic, it is a
contradiction and we have uniqueness of equilibrium state. ∎

##  4\. Proof of Corollary 1

In order to prove Corollary 1, we observe that both the sets $\mathcal{ZP}$
and $\mathcal{HP}$ of random zooming and random hyperbolic potentials,
respectively, are open. Since we have that $\mathcal{HP}\subset\mathcal{ZP}$,
we are done if we show that both open sets have same closure.

We begin by observing that the closure of the random hyperbolic potentials is
the set of potentials $\phi\in\mathbb{L}^{1}_{\mathbb{P}}(\Omega,C^{0}(M))$
such that the following inequality holds:

(6) $P_{F|T}(\phi,\Lambda^{c})\leq P_{F|T}(\phi,\Lambda)=P_{F|T}(\phi).$

In this case, for the potential $\phi:\Omega\times M\to\mathbb{R}$, we can
also obtain a sequence of measures $\mu_{k}\in\mathcal{Z}$ such that

$h_{\mu_{k}}(F|T)+\int\phi d\mu_{k}\to P(\phi).$

It also provides some equilibrium state $\mu_{0}$ which is a random zooming
measure. So, we have the equality:

$P(\phi)=h_{\mu_{0}}(F|T)+\int\phi d\mu_{0}.$

It implies that the potential $\phi$ is in the closure of the set
$\mathcal{ZP}$ and $\overline{\mathcal{ZP}}\subset\overline{\mathcal{HP}}$.
Since we have that $\overline{\mathcal{HP}}\subset\overline{\mathcal{ZP}}$, we
obtain $\overline{\mathcal{HP}}=\overline{\mathcal{ZP}}$ and the proof is
done.

##  5\. Examples

In this section, we give examples of random zooming systems, as the random
non-uniformly expanding maps so-called random Viana maps and construct zooming
potentials. Random non-uniformly expanding maps are examples of random zooming
systems when the contractions are exponential. In this case, random zooming
times are called random hyperbolic times.

We begin by constructing random zooming potentials for random maps which has
zooming fixed points. The Dirac probability supported on this point will be
the equilibrium state. After that, we show that the null potentials is random
zooming, if we can find a potential with an equlibrium state which is not a
measure of maximal entropy at the same time. Finally, we presente the class of
random Viana maps and give an example of random zooming system which does not
have exponential zooming contractions.

###  5.1. Random Zooming Potentials

Let $F$ be an random zooming map with a critical set such that the random
zooming set $\Lambda$ contains a fixed point $x_{0}$. We will construct a
random zooming potential, that is, a potential $\varphi$ such that

$\sup_{\nu\in\mathcal{Z}^{c}}\bigg{\\{}h_{\nu}(F|T)+\int\varphi
d\nu\bigg{\\}}<\sup_{\mu\in\mathcal{Z}}\bigg{\\{}h_{\mu}(F|T)+\int\varphi
d\mu\bigg{\\}},$

where $\mathcal{Z}$ is the set of the zooming measures.

Since $x_{0}\in\Lambda$ is a fixed point, the Dirac probability
$\delta_{x_{0}}$ at $x_{0}$ is an invariant zooming measure.

Let $\phi\geq 0$ be a continuous potential on the fibers such that
$\max\phi=\phi(x_{0})>0$ and it is integrable with respect to the marginal
probability (for example, we can define it as constant on the fibers). We have
that

$\int\phi d\nu<\phi(x_{0})=\int\phi
d\delta_{x_{0}},\forall\nu\in\mathcal{Z}^{c}.$

We assume that

$\sup_{\nu\in\mathcal{Z}^{c}}\bigg{\\{}\int\phi d\nu\bigg{\\}}<\phi(x_{0}).$

So, we can find $k>0$ such that

$\int k\phi d\delta_{x_{0}}-\sup_{\nu\in\mathcal{Z}^{c}}\bigg{\\{}\int k\phi
d\nu\bigg{\\}}>2h(F)\implies\int k\phi d\delta_{x_{0}}-\int k\phi
d\nu>2h(F),\forall\nu\in\mathcal{Z}^{c}$

where $h(F)>0$ is the topological entropy. It implies that

$h_{\nu}(F|T)+\int k\phi d\nu\leq h(F)+\int k\phi d\nu<2h(F)+\int k\phi
d\nu<\int k\phi d\delta_{x_{0}},$

that is,

$\sup_{\nu\in\mathcal{Z}^{c}}\bigg{\\{}h_{\nu}(F|T)+\int k\phi
d\nu\bigg{\\}}<\int k\phi
d\delta_{x_{0}}\leq\sup_{\mu\in\mathcal{Z}}\bigg{\\{}h_{\mu}(F|T)+\int k\phi
d\mu\bigg{\\}}.$

Then, the potential $\varphi=k\phi$ is random zoooming.

Also, in general, if there exist a potential $\phi$ and a measure
$\mu\in\mathcal{Z}$, such that

$\sup_{\nu\in\mathcal{Z}^{c}}\bigg{\\{}\int\phi d\nu\bigg{\\}}<\int\phi d\mu,$

we can find $k>0$ such that

$\int k\phi d\mu-\sup_{\nu\in\mathcal{Z}^{c}}\bigg{\\{}\int k\phi
d\nu\bigg{\\}}>2h(F)\implies\int k\phi d\mu-\int k\phi
d\nu>2h(F),\forall\nu\in\mathcal{Z}^{c}$

where $h(F)>0$ is the topological entropy. It implies that

$h_{\nu}(F|T)+\int k\phi d\nu\leq h(F)+\int k\phi d\nu<2h(F)+\int k\phi
d\nu<\int k\phi d\mu,$

that is,

$\sup_{\nu\in\mathcal{Z}^{c}}\bigg{\\{}h_{\nu}(F|T)+\int k\phi
d\nu\bigg{\\}}<\int k\phi
d\mu\leq\sup_{\mu\in\mathcal{Z}}\bigg{\\{}h_{\mu}(F|T)+\int k\phi
d\mu\bigg{\\}}.$

Then, the potential $\varphi=k\phi$ is random zooming.

###  5.2. The Null Potential

We will show that for random maps and potentials for which there exist
equilibrium states we have that the null potential is random zooming and, as a
consequence, potentials with low variation are also random zooming. It is a
consequence of the fact that the set of random zooming potentials is open.

In order to show it, we use the existence of equilibrium states which are
zooming measures.

We recall that $\mathcal{Z}$ denotes the set of zooming invariant measures.
Assume, by contradiction, that

$\sup_{\mu\in\mathcal{Z}}\\{h_{\mu}(F|T)\\}\leq\sup_{\nu\in\mathcal{Z}^{c}}\\{h_{\nu}(F|T)\\}.$

Let $\phi_{0}$ be a random zooming potential which has equilibrium states.
There exists an ergodic equilibrium state $\mu\in\mathcal{Z}$. We have that
$\mu$ is also an equilibium state for $\phi:=\phi_{0}-P(\phi_{0})$ which
implies that $P(\phi)=0$ and $h_{\mu}(F|T)+\int\phi d\mu=0$ and
$-h_{\mu}(F|T)=\int\phi d\mu\leq 0$. Assume that there exists
$\nu_{0}\in\mathcal{Z}^{c}$ such that $h_{\mu}(F|T)<h_{\nu_{0}}(F|T)$. Since
every equilibrium state is a zooming probability, we have that $\nu_{0}$ is
not an equilibrium state for $\phi$ and

$h_{\nu_{0}}(F|T)+\int\phi d\nu_{0}<h_{\mu}(F|T)+\int\phi d\mu=0,$

which implies that $\int\phi d\nu_{0}<\int\phi d\mu=-h_{\mu}(F|T)$ because
$h_{\mu}(F|T)<h_{\nu_{0}}(F|T)$. For every $t\in(0,1)$ and putting
$\nu=(1-t)\mu+t\nu_{0}$ we still have that $h_{\mu}(F|T)<h_{\nu}(F|T)$ (we
observe that $h_{\nu}(F|T)=(1-t)h_{\mu}(F|T)+th_{\nu_{0}}(F|T)$, because the
entropy is an affine function, see [16, Proposition 9.6.1] for the
deterministic case and by using the Abramov-Rokhlin Formula we obtain it for
the relative entropy). Let $\epsilon>0$ and $t\in(0,1),n\in\mathbb{N}$ are
chosen such that such that $0<h_{\nu}(F|T)-h_{\mu}(F|T)-\epsilon/n$ and

$\int\phi d\mu>\int\phi d\nu>\int\phi
d\mu-\epsilon\implies-\epsilon(n-1)/n<h_{\nu}(F|T)-h_{\mu}(F|T)-\epsilon<h_{\nu}(F|T)+\int\phi
d\nu<0,$

using that $\int\phi d\mu=-h_{\mu}(F|T)$ . It is possible because once $\mu$
is an equilibrium state we have $h_{\nu_{0}}(F|T)-h_{\mu}(F|T)<\int\phi
d\mu-\int\phi d\nu_{0}$ and there exist $t\in(0,1),n\in\mathbb{N}$ such that

$\epsilon/n<h_{\nu}(F|T)-h_{\mu}(F|T)=t(h_{\nu_{0}}(F|T)-h_{\mu}(F|T))<t\bigg{(}\int\phi
d\mu-\int\phi d\nu_{0}\bigg{)}=\int\phi d\mu-\int\phi d\nu<\epsilon.$

We obtain a contradiction because $\nu\in\mathcal{Z}^{c}$ and $\epsilon>0$ is
arbitrary and the potential $\phi$ is zooming. It shows that we have

$\sup_{\nu\in\mathcal{Z}^{c}}\\{h_{\nu}(F|T)\\}\leq h_{\mu}(F|T).$

If $\mu_{0}$ is an equilibrium state which is not a measure of maximal
entropy, then we have

$\sup_{\nu\in\mathcal{Z}^{c}}\\{h_{\nu}(F|T)\\}\leq
h_{\mu_{0}}(F|T)<\sup_{\mu\in\mathcal{Z}}\\{h_{\mu}(F|T)\\}=h(f).$

Hence, the null potential is random zooming.

###  5.3. Random Viana maps

Deterministic Viana maps are presented in [1, 24]. Alves and Araújo in [2],
considering $f_{0}$ as Viana map and the functions $f\in C^{2}$ around of
Viana map must have the same critical point set $\mathcal{C}$ and impose that

$Df_{t}(x)=Df_{0}(x)\quad\text{for every}\quad x\in M\setminus\mathcal{C}\ \
\text{and}\ \ t\in Y,$

where $Y$ metric space with probability measure $\nu$. They also consider
that, in the case of maps with critical sets, they assume a slow approximation
of random orbits to the critical set: given any small $\gamma>0$ there is
$\delta>0$ such that

$\limsup_{n\to+\infty}\dfrac{1}{n}\sum_{j=0}^{n-1}-\log
dist_{\delta}(f_{\underline{t}}^{j}(x),\mathcal{C})\leq\gamma$

for $\nu^{\mathbb{N}}\times m$ almost every $(\underline{t},x)\in\Omega\times
M$ for $\Omega=Y^{\mathbb{N}}$. Where $dist_{\delta}(x,\mathcal{C})$ is
$\delta-$truncated distance from $x\in M$ to $\mathcal{C}$,
$\underline{t}=(t_{1},t_{2},\dots)\in\Omega$ and the random orbit
$(f_{\underline{t}}^{n}(x))_{n\geq 1}$ defined

$f_{\underline{t}}^{n}=f_{t_{n}}\circ\cdots\circ f_{t_{1}}\quad\text{for}\quad
n\geq 1.$

Now, under these conditions, they show that an RDS non-uniformly expanding
with critical points is obtained, through the Proposition 2.6 (see [2], pag
35), while at the Proposition 2.3 (see [2], pag 32) the existence of
hyperbolic times is obtained.
On the other hand, considering $T:\Omega\rightarrow\Omega$ the left shift map
and $\nu^{\mathbb{N}}$ ergodic product measure on $(T,\Omega)$, then we have

  1. i)

A measure $\mu\in\mathcal{Z}$ which is random zooming: since the deterministic
Viana map has an expanding fixed point and the shift map also has fixed
points, as can obtain a fixed point for the random Viana map which is a
zooming point. Also, the Dirac probability supported on this fixed zooming
point is a random zooming measure;

  2. ii)

A potential $\varphi:\Omega\times M\rightarrow\mathbb{R}$ for the random Viana
map which is a zooming potential (see 4). In the next subsection we construct
such a potential which is random zooming, for any random map which has a
zooming fixed point.

  3. iii)

The null potential is random zooming and we have the existence of measures of
maximal entropy.

###  5.4. Random Zooming but not Expanding

We can obtain an example of a random zooming system which does not have
exponential contractions by considering the product of the deterministic
example given in [18] and the shift map of finite type. We obtain an skew
product with contractions on the fibers which are not exponential. We refer
[18], Example 9.14. We can see that for this example we have the contractions
as

$\alpha_{n}(r)=\bigg{(}\frac{1}{1+n\sqrt{r}}\bigg{)}^{2}r.$

For the random system, we consider it between fibers.

## References

  * [1] J. F. Alves, Statistical Analysis of Non-Uniformly Expanding Dynamical Systems, $24^{\circ}$ Colóquio Brasileiro de Matemática, 2003.
  * [2] J. F. Alves and V. Araújo, Random perturbations of nonuniformly expanding maps.  Astérisque, 286; 25 - 62, 2003.
  * [3] J. F. Alves, K. Oliveira, E. Santana, Equilibrium States for Non-Uniformly Expanding Maps with Critical Sets, arXiv: 2003.11620
  * [4] A. Arbieto, C. Matheus, K. Oliveira, Equilibrium States for Random Non-Uniformly Expanding Maps, Nonlinearity, 17, 2004, 581-593.
  * [5] R. A. Bilbao, V. Ramos, Uniqueness and stability of equilibrium states for random non-uniformly expanding maps, Ergodic Theory and Dynamical Systems, 43, 2023, 2589 - 2623.
  * [6] H. Bruin, G. Keller, Equilibrium States for S-Unimodal Maps, Ergodic Theory and Dynamical Systems, 18, 1998, 765-789.
  * [7] J. Buzzi, O. Sarig, Uniqueness of Equilibrium Measures for Countable Markov Shifts and Multidimensional Piecewise Expanding Maps, Ergodic Theory and Dynamamical Systems, 23, 2003, 1383-1400.
  * [8] M. Denker, M. Urbanski, Ergodic Theory of Equilibrium States for Rational Maps, Nonlinearity, 4, 1991, 103-134.
  * [9] G. Iommi, M. Todd, Thermodynamic Formalism for Interval Maps: Inducing Schemes, Dynamical Systems, 28, 3, 2013, 354-380.
  * [10] G. Iommi, M. Todd, Natural Equilibrium States for Multimodal Maps, Communications in Mathematical Physics, 300, 2010, 65-94.
  * [11] Yu. Kifer, Thermodynamic formalism for random transformations revisited, Stochastics and Dynamics, 81: 77–102, 2008.
  * [12] P. D. Liu, Dynamics of random transformations: smooth ergodic theory, Erg. Theory and Dyn. Systems. 21: 1279–1319, 2001.
  * [13] H. Li, J. Rivera-Letelier, Equilibrium States of Weakly Hyperbolic One-Dimensional Maps for Hölder Potentials, Communications in Mathematical Physics, 328, 2014, 397-419.
  * [14] R. Leplaideur, K. Oliveira, I. Rios, Invariant Manifolds and Equilibrium States for Non-Uniformly Hyperbolic Horseshoes, Nonlinearity, 19, 2006, 2667-2694.
  * [15] K. Oliveira, Equilibrium States for Non-Uniformly Expanding Maps, Ergodic Theory and Dynamical Systems, 23, 06, 2003, 1891-1905.
  * [16] K. Oliveira, M. Viana, Foundations of Ergodic Theory, Cambridge University Press, 2016.
  * [17] Ya. Pesin, Dimension Theory in Dynamical Systems: Contemporary Views and Applications, The University of Chicago Press, 1997.
  * [18] V. Pinheiro, Expanding Measures, Annales de l’Institute Henri Poincaré, 28, 2011, 889-939.
  * [19] V. Pinheiro, P. Varandas, Thermodynamic Formalism for Expanding Measures, arXiv:2202.05019
  * [20] V. Ramos, M. Viana, Equilibrium States for Hyperbolic Potentials, Nonlinearity, 30, 2017, 825-847.
  * [21] E. Santana, Equilibrium States for Open Zooming Systems, arXiv:2010.08143
  * [22] M. Stadlbauer, S. Suzuki and P. Varandas.Thermodynamic formalism for random non-uniformly expanding maps. Commun. Math. Phys. 385: 369–427, 2021.
  * [23] P. Varandas, M. Viana, Existence, Uniqueness and Stability of Equilibrium States for Non-Uniformly Expanding Maps, Annales de l’Institute Henri Poincaré, 27, 2010, 555-593.
  * [24] M. Viana, Multidimensional NonHyperbolic Attractors, Publications Mathématiques de l’Institut des Hautes Études Scientifiques, 85, 1, 1997, 63-96. American Journal of Mathematics, 97, 1975, 937-997.
  * [25] M. Yuri, Thermodynamical Formalism for Countable to One Markov Systems, Transactions of the American Mathematical Society, 335, 2003, 2949-2971.


# UNCOVER: JWST Spectroscopy of Three Cold Brown Dwarfs at Kiloparsec-scale
Distances

Adam J. Burgasser Department of Astronomy & Astrophysics, UC San Diego, La
Jolla, CA 92093, USA  Rachel Bezanson Department of Physics and Astronomy and
PITT PACC, University of Pittsburgh, Pittsburgh, PA 15260, USA  Ivo Labbe
Centre for Astrophysics and Supercomputing, Swinburne University of
Technology, Melbourne, VIC 3122, Australia  Gabriel Brammer Cosmic Dawn Center
(DAWN), Niels Bohr Institute, University of Copenhagen, Jagtvej 128, København
N, DK-2200, Denmark  Sam E. Cutler Department of Astronomy, University of
Massachusetts, Amherst, MA 01003, USA  Lukas J. Furtak Physics Department,
Ben-Gurion University of the Negev, P.O. Box 653, Be’er-Sheva 84105, Israel
Jenny E. Greene Department of Astrophysical Sciences, Princeton University, 4
Ivy Lane, Princeton, NJ 08544  Roman Gerasimov Department of Physics, UC San
Diego, La Jolla, CA 92093, USA  Department of Physics & Astronomy, University
of Notre Dame, Notre Dame, IN 46556, USA  Joel Leja Department of Astronomy &
Astrophysics, The Pennsylvania State University, University Park, PA 16802,
USA  Institute for Computational & Data Sciences, The Pennsylvania State
University, University Park, PA 16802, USA  Institute for Gravitation and the
Cosmos, The Pennsylvania State University, University Park, PA 16802, USA
Richard Pan Department of Physics and Astronomy, Tufts University, 574 Boston
Ave., Medford, MA 02155, USA  Sedona H. Price Department of Physics and
Astronomy and PITT PACC, University of Pittsburgh, Pittsburgh, PA 15260, USA
Bingjie Wang (王冰洁) Department of Astronomy & Astrophysics, The Pennsylvania
State University, University Park, PA 16802, USA  Institute for Computational
& Data Sciences, The Pennsylvania State University, University Park, PA 16802,
USA  Institute for Gravitation and the Cosmos, The Pennsylvania State
University, University Park, PA 16802, USA  John R. Weaver Department of
Astronomy, University of Massachusetts, Amherst, MA 01003, USA  Katherine E.
Whitaker Department of Astronomy, University of Massachusetts, Amherst, MA
01003, USA  Cosmic Dawn Center (DAWN), Denmark  Seiji Fujimoto Hubble Fellow
Department of Astronomy, The University of Texas at Austin, Austin, TX 78712,
USA  Vasily Kokorev Kapteyn Astronomical Institute, University of Groningen,
9700 AV Groningen, The Netherlands  Pratika Dayal Kapteyn Astronomical
Institute, University of Groningen, 9700 AV Groningen, The Netherlands
Themiya Nanayakkara Centre for Astrophysics and Supercomputing, Swinburne
University of Technology, PO Box 218, Hawthorn, VIC 3122, Australia  Christina
C. Williams NSF’s National Optical-Infrared Astronomy Research Laboratory, 950
N. Cherry Avenue, Tucson, AZ 85719, USA  Danilo Marchesini Department of
Physics & Astronomy, Tufts University, MA 02155, USA  Adi Zitrin Physics
Department, Ben-Gurion University of the Negev, P.O. Box 653, Be’er-Sheva
84105, Israel  Pieter van Dokkum Astronomy Department, Yale University, 52
Hillhouse Ave, New Haven, CT 06511, USA

(Accepted 22 Nov 2023)

###### Abstract

We report JWST/NIRSpec spectra of three distant T-type brown dwarfs identified
in the Ultradeep NIRSpec and NIRCam ObserVations before the Epoch of
Reionization (UNCOVER) survey of the Abell 2744 lensing field. One source was
previously reported as a candidate T dwarf on the basis of NIRCam photometry,
while two sources were initially identified as candidate active galactic
nuclei. Low-resolution 1–5 $\micron$ spectra confirm the presence of molecular
features consistent with T dwarf atmospheres, and comparison to spectral
standards infers classifications of sdT1, T6, and T8–T9. The warmest source,
UNCOVER-BD-1, shows evidence of subsolar metallicity, and atmosphere model
fits indicates Teff = 1300 K and [M/H] $\sim$ $-$1.0, making this one of the
few spectroscopically-confirmed T subdwarfs known. The coldest source,
UNCOVER-BD-3, is near the T/Y dwarf boundary with Teff = 550 K, and our
analysis indicates the presence of PH3 in the 3–5 $\micron$ region, favored
over CO2 and a possible indicator of subsolar metallicity. We estimate
distances of 0.9–4.5 kpc from the Galactic midplane, making these the most
distant brown dwarfs with spectroscopic confirmation. Population simulations
indicate high probabilities of membership in the Galactic thick disk for two
of these brown dwarfs, and potential halo membership for UNCOVER-BD-1. Our
simulations indicate that there are approximately 5 T dwarfs and 1–2 L dwarfs
in the Abell 2744 field down to F444W = 30 AB mag, roughly one-third of which
are thick disk members. These results highlight the utility of deep
JWST/NIRSpec spectroscopy for identifying and characterizing the oldest metal-
poor brown dwarfs in the Milky Way.

Brown dwarfs (185), T dwarfs (1679), T subdwarfs (1680), Milky Way stellar
halo (1060), Sky surveys (1464)

††facilities: JWST(NIRCam), JWST(NIRSpec), HST(ACS), HST(WFC3)††software:
astropy (Astropy Collaboration et al., 2013, 2018, 2022), msaexp (Brammer,
2022), JWST Calibration pipeline (Bushouse et al., 2023), SPLAT (Burgasser &
Splat Development Team, 2017),

##  1 Introduction

Brown dwarfs are stellar objects with masses below the $\sim$0.075 M⊙
threshold for sustained core hydrogen fusion (Kumar, 1962, 1963; Hayashi &
Nakano, 1963). Supported by electron degeneracy pressure, these compact,
hydrogen-rich objects radiate their initial heat of formation and continuously
cool over time. The relatively high abundance of brown dwarfs in the immediate
vicinity of the Sun ($\gtrsim$20% of stars; Kirkpatrick et al. 2021; Reylé et
al. 2021), their lack of chemical processing, and their time-dependent
luminosities make them useful probes of the ages and chemical evolution of
various Milky Way populations (Stauffer et al., 1998; Gerasimov et al., 2022).
However, the intrinsic faintness of cool brown dwarfs largely limits their
detection to the immediate Solar Neighborhood ($d\lesssim$ 100 pc).

The unprecedented sensitivity of JWST at near-infrared wavelengths where brown
dwarfs are brightest enables detection of brown dwarfs deep into the Galactic
halo (Ryan & Reid, 2016; Aganze et al., 2022). Multiple brown dwarf candidates
have already been identified in deep multi-band imaging surveys (Nonino et
al., 2023; Glazebrook et al., 2023; Wang et al., 2023b; Hainline et al., 2023;
Holwerda et al., 2023). However, confirmation and physical characterization of
these distant brown dwarfs, including accurate measurement of temperatures,
surface gravities, and compositions, requires spectroscopy.

The JWST Cycle 1 Treasury program Ultradeep NIRSpec and NIRCam ObserVations
before the Epoch of Reionization (UNCOVER; Bezanson et al. 2022) aims to
identify some of the highest redshift and faintest Epoch of Reionization (EoR)
systems that are lensed by the massive z=0.308 galaxy cluster Abell 2744.
UNCOVER combines deep JWST NIRCam imaging over $\sim 45$ arcmin2 in seven
filters (F115W, F150W, F200W, F277W, F356W, F410M, F444W) down to $\sim 30$ AB
mag (Weaver et al., 2023) with NIRSpec PRISM 0.6–5 $\micron$ low-resolution
spectroscopy. To date, UNCOVER has photometrically identified and
characterized $\sim 50,000$ galaxies in the Abell 2744 field, and has
spectroscopically confirmed galaxies out to $z\gtrsim 10$ (e.g., Wang et al.
2023a), characterized the hosts and supermassive black holes of multiple high-
redshift active galactic nuclei (AGN; e.g., Goulding et al. 2023; Furtak et
al. 2023), and performed the first spectroscopic analysis of low luminosity
EoR galaxies (Atek et al., 2023).

In this article, we utilize the deep imaging and spectroscopy of UNCOVER to
identify and spectroscopically characterize three T-type brown dwarfs. Section
2 summarizes the identification and observations of these sources. Section 3
describes our analysis of JWST/NIRSpec spectra, including classification,
model fitting, and estimation of the distances by scaling model surface
fluxes. Section 4 presents population simulations that justifies the detection
of three T dwarfs in the narrow Abell 2744 field, and provides statistical
constraints on the Galactic population membership. Section 5 summarizes our
results. We note that these sources and analysis of the data presented here
have also been reported contemporaneously by Langeroodi & Hjorth (2023).

##  2 Identification and Observations

Catalog designations, coordinates, and AB magnitudes from Weaver et al. (2023)
of the three sources presented here are listed in Table 2. Two of the sources
were initially identified as candidate high-redshift AGN in UNCOVER NIRCam
photometry (Labbe et al., 2023). The third source, 39243, was previously
identified as the candidate T dwarf GLASS-BD-1 (aka “Nonino’s Dwarf”) based on
multi-band NIRCam photometry from the Grism Lens-Amplified Survey from Space
(GLASS) survey (Nonino et al., 2023; Glazebrook et al., 2023). All three
sources were included in the first UNCOVER NIRSpec campaign.

NIRSpec/PRISM observations were obtained on 31 July – 2 August 2023 (UT),
split over seven Multi-shutter array (MSA) masks. All observations were taken
with a 2-POINT-WITH-NIRCam-SIZE2 dither pattern and a three shutter slitlet
nod pattern at an angle of $\mathrm{V3PA\sim 266^{\circ}}$. Sources 32265 and
33437 were observed on MSAs 3, 5, 6, and 7, using the NRSIRS2RAPID readout
pattern for MSA 3 and NRSIRS2 for the rest, for a total of 14.3 hours. Source
ID 39243 was observed on MSA 4 with readout pattern NRSIRS2 for a total of 4.4
hours. Full observational details will be presented in Price et al. (2023, in
prep).

The spectra were reduced using msaexp (v0.6.10; Brammer 2022), starting from
the level 2 data products downloaded from the Mikulski Archive for Space
Telescopes (MAST)11110.17909/8k5c-xr27 (catalog 10.17909/8k5c-xr27), reduced
using version 1.11.3 of the JWST Calibration pipeline (Bushouse et al., 2023).
For each slit mask, msaexp corrects $1/f$-noise, finds and masks cosmic ray
“snowballs,” and removes the bias in each individual frame. It then applies a
world coordinate system, identifies each slit, and applies flat-fielding and
photometric corrections. The 2D slits are combined and drizzled onto a common
grid, a local background subtraction is applied, and spectra are then
optimally extracted (Horne, 1986). For the early spectroscopic reduction
presented here (internal v0.3), flux calibration was modeled as a first-order
polynomial determined by convolving the individual mask 1D spectra with the
broad/medium band filters, and comparing to the total photometry (Weaver et
al., 2023).

Figure 1 shows the calibrated JWST/NIRSpec spectra over the 1–5 $\mu$m band.
In the 1–2.5 $\mu$m near-infrared region, we identify the 1.0, 1.25, 1.6, and
2.1 $\mu$m flux peaks characteristic of cool brown dwarfs, shaped by strong
H2O and CH4 bands (Burgasser et al., 2006b). This is followed by a broad
minimum spanning 2.3–3.5 $\mu$m which encompasses H2O, CH4, and collision-
induced H2 absorption (Leggett et al., 2015; Linsky, 1969). A red peak emerges
between 3.5 and 4.5 $\mu$m, showing the most structure for Source 39243. This
region is shaped by overlapping bands of H2O, CH4, NH3, CO2, CO, and possibly
PH3 in cool brown dwarfs (Yamamura et al., 2010; Sorahana & Yamamura, 2012;
Leggett et al., 2015; Miles et al., 2020; Beiler et al., 2023), as discussed
further below. We conclude from these spectral morphologies that all three
sources are low-temperature brown dwarfs, and are hereafter designated as
UNCOVER-BD sources.

Figure 1: JWST/NIRSpec PRISM spectra of the three brown dwarf candidates
observed by UNCOVER in $F_{\lambda}$ units (black lines with grey shaded
uncertainties), normalized in the 1.1–1.3 $\mu$m region to a maximum value of
0.95. Each spectrum is offset by a constant (dashed lines), and major spectral
features in the 1–5 $\micron$ region are labeled. A bad pixel at 1.8 $\mu$m
has been masked out in the spectrum of UNCOVER-BD-1.  Table 1: Properties of
the UNCOVER T Dwarfs

Property | UNCOVER-BD-1 | UNCOVER-BD-2 | UNCOVER-BD-3aaaka GLASS-BD-1 or ”Nonino’s Dwarf” (Nonino et al., 2023). | Ref
---|---|---|---|---
HST & JWST Astrometry & PhotometrybbPhotometry in AB magnitudes from Weaver et
al. (2023) (internal version 3.0.0).
Coordinate | J00140901-3022126 | J00141114-3021585 | J00140333-3021217 | [1]
MSA IDccMulti-Slit Array identification number (Weaver et al., 2023, internal version 2.2.1). | 32265 | 33437 | 39243 | [1]
F814W (AB) | 27.38$\pm$0.31 |  $\cdots$ |  $\cdots$ | [1]
F115W (AB) | 28.26$\pm$0.11 | 27.34$\pm$0.06 | 28.00$\pm$0.06 | [1]
F125W (AB) |  $\cdots$ |  $\cdots$ | 27.57$\pm$0.49 | [1]
F150W (AB) | 28.06$\pm$0.09 | 28.10$\pm$0.10 | 28.81$\pm$0.18 | [1]
F160W (AB) |  $\cdots$ |  $\cdots$ | 27.49$\pm$0.69 | [1]
F200W (AB) | 28.04$\pm$0.10 | 28.87$\pm$0.19 | 29.71$\pm$0.35 | [1]
F277W (AB) | 28.25$\pm$0.08 | 29.03$\pm$0.16 | 29.16$\pm$0.14 | [1]
F356W (AB) | 27.57$\pm$0.03 | 28.06$\pm$0.06 | 27.69$\pm$0.03 | [1]
F410M (AB) | 27.04$\pm$0.04 | 26.71$\pm$0.03 |  $\cdots$ | [1]
F444W (AB) | 27.28$\pm$0.04 | 26.97$\pm$0.03 | 25.637$\pm$0.005 | [1]
Spectral Properties
Integration Time (hr) | 14.3 | 14.3 | 4.4 | [2]
Spectral Type | sdT1 | T6 | T8–T9 | [2]
TeffddSingle values based on best fits to LOWZ models, while ranges for Teff and $\log{g}$ encapsulate the fits to all five models examined. (K) | 1300 (1100–1500) | 1000 (1000-1100) | 550 (500–700) | [2]
$\log{g}$ddSingle values based on best fits to LOWZ models, while ranges for Teff and $\log{g}$ encapsulate the fits to all five models examined. (cgs) | 5.0 (4.5–5.0) | 5.25 (5.25-5.5) | 5.25 (5.0–5.5) | [2]
$[M/H]$ddSingle values based on best fits to LOWZ models, while ranges for Teff and $\log{g}$ encapsulate the fits to all five models examined. | -1.0 | 0.0 | -0.5 | [2]
Other Inferred Properties
dmodeleeBased on scaling surface fluxes of best model fits to observed apparent spectral fluxes and assuming a radius of 1 Jupiter radius. (kpc) | 4.5$\pm$1.2 | 2.32$\pm$0.23 | 0.87$\pm$0.30 | [2]
$\mathcal{P}$(thin disk) |  $<$1% | 16% | 75% | [2]
$\mathcal{P}$(thick disk) | 76% | 79% | 25% | [2]
$\mathcal{P}$(halo) | 24% | 6% |  $<$1% | [2]

References. — [1] Weaver et al. (2023); [2] This paper.

##  3 Spectral Analysis

We classified the sources by comparing their 1–2.4 $\mu$m spectra to L and T
dwarf spectral standards from the SpeX Prism Library Analysis Toolkit (SPLAT;
Burgasser & Splat Development Team 2017). We identified the standard with the
smallest reduced $\chi^{2}_{r}$ residual,222We compute chi-square as
$\chi^{2}\equiv\sum_{i}\frac{(O_{i}-\alpha{C_{i}})^{2}}{\sigma_{i}^{2}}$,
where $\mathbf{O}$ is the observed spectrum, $\mathbf{C}$ the comparison
template, $\mathbf{\sigma}$ the observed spectrum uncertainties, and
$\alpha=\frac{\sum_{i}O_{i}C_{i}/\sigma_{i}^{2}}{\sum_{i}C_{i}^{2}/\sigma_{i}^{2}}$
is a scaling factor that minimizes $\chi^{2}$. We compute reduced chi-square
as $\chi^{2}_{r}\equiv\chi^{2}/(N-1)$, where N is the number of spectral data
points. shown in the left panels of Figure 2. For UNCOVER-BD-1, the T1
standard provides the best overall fit, but has clear deviations in each of
the $J$-, $H$-, and $K$-band spectral peaks. In particular, UNCOVER-BD-1 shows
excess flux on the blue side of the $H$-band while having a relatively
suppressed $K$-band. As discussed below, we infer these deviations are caused
by subsolar metallicity. UNCOVER-BD-2 is an excellent match to the T6
standard, while UNCOVER-BD-3 equally matches T8 and T9 standards, likely due
to the spectrum’s lower signal-to-noise.

Figure 2: (Left panels): Comparison of the 1-2.4 $\micron$ JWST/NIRSpec PRISM
spectra (black lines with grey shaded uncertainties in apparent flux density
units) to their best-fit near-infrared spectral standards from SPLAT (magenta
lines). Standard spectra are from Burgasser et al. (2004, 2006a). (Right
panels): Best-fits LOWZ models to the full 1–5 $\micron$ spectra (blue lines),
with parameters indicated in the legends.

To quantify the physical properties of these sources, we compared the
JWST/NIRSpec spectra to five sets of low-temperature atmosphere models: BT-
Settl (Allard et al., 2012), ATMO (Phillips et al., 2020), Sonora-Bobcat
(Marley et al., 2021), Sonora-Cholla (Karalidi et al., 2021), and LOWZ
(Meisner et al., 2021). These models span the temperature range of T- and
Y-type dwarfs (roughly 500–1500 K), with varying parameters and treatments for
metal composition, opacities, chemical equilibrium, and cloud formation. We
focused on models with surface gravities $\log{g}\geq 4$ (cgs), consistent
with ages $\gtrsim$100 Myr. We smoothed and interpolated each model to match
the resolution of the NIRSpec/PRISM data over the 1–5 $\mu$m band, scaling
model fluxes to minimize $\chi^{2}_{r}$ residuals. We performed a simple grid
fit, identifying the lowest $\chi^{2}_{r}$ match to the data within each grid.

Of the five model sets, the LOWZ models consistently provided the best fits
(Figure 2), although all five models gave equivalent values for temperature
and surface gravity. The LOWZ models are also computed at subsolar
metallicities, and it is notable that both UNCOVER-BD-1 and UNCOVER-BD-3 are
best fit to subsolar metallicity models. Our inferred temperature for UNCOVER-
BD-3, 500 K $\lesssim$ Teff $\lesssim$ 700 K, is also consistent with the 650
K estimate by Nonino et al. (2023) based on NIRCam photometry. Finally, we
note that CO, CH4, and NH3 abundances are strongly affected by non-equilibrium
mixing in brown dwarf atmospheres, and all three molecules exhibit absorption
features in the 1–5 $\mu$m range (Figure 1). The best-fit ATMOS and Sonora-
Cholla models both indicate enhanced vertical mixing with $\log\kappa_{zz}$ =
7–8 (cgs) for UNCOVER-BD-1, while UNCOVER-BD-2 and UNCOVER-BD-3 are best
matched to low $\kappa_{zz}$/chemical equilibrium models.

The atmosphere models are computed as surface fluxes; hence, our fits to
apparent flux densities constraint the scale factor $\alpha=(R/d)^{2}$, where
$R$ is the radius of the source and $d$ its distance. Assuming a common radius
of 1 Jupiter radius for all three brown dwarfs, averaging over all best-fit
models, and ignoring the presence of binary companions, we infer the
spectroscopic distance estimates listed in Table 2. Our distance estimate of
870$\pm$300 pc for UNCOVER-BD-3 overlaps with the 570-720 pc estimate by
Nonino et al. (2023), while the other two brown dwarfs have estimated
distances exceeding 2 kpc. To our knowledge, these are the most distant T
dwarfs to have measured spectroscopy.

##  4 Population Constraints

As the Abell 2744 field is at a high Galactic latitude ($b$ = $-81\arcdeg$),
the kpc-scale distances of these brown dwarfs translate into large vertical
offsets from the Galactic plane, and hence a high likelihood of being members
of the thick disk or halo populations. To assess population membership and to
ascertain whether the number of T dwarfs found is consistent with
expectations, we constructed a Galactic population simulation of thin disk,
thick disk, and halo low-mass stars and brown dwarfs, accounting for
substellar evolution and population spatial distributions. Full details of the
simulation are provided in the Appendix.

Figure 3: (Left) Cumulative number counts of expected L and T dwarfs in the
UNCOVER field of view as a function of limiting F444W magnitude. Sources are
required to be detected ($<$30 AB mag) in at least three filters. Counts are
broken down by spectral class (L dwarfs in blue, T dwarfs in magenta, total in
black) and by population (thin disk as solid lines, thick disk as dashed
lines, halo as dotted lines, total as thick solid lines). The grey horizontal
line and band delineates the number of confirmed T dwarfs in the Abell 2744
field with Poisson uncertainties. (Right): Relative fraction of thin disk
(blue line), thick disk (orange line), and halo (green line) as a function of
distance for the T dwarfs in our simulation. The vertical regions indicate the
estimated distances and 1$\sigma$ uncertainties of the three T dwarfs reported
here.

Figure 3 shows the overall predicted numbers of detectable ultracool dwarfs in
the Abell 2744 field based on these simulations, with the numbers of late-M,
L, and T dwarfs disaggregated. At magnitudes F444W $\gtrsim$ 26 AB, T dwarfs
are the dominant population of ultracool dwarfs in this field, and are
primarily thin disk sources. By F444W = 30 AB, our simulations predict
approximately 5 T dwarfs in the Abell 2744 field, $\approx$60% from the thin
disk, $\approx$35% from the thick disk, and $\lesssim$5% from the halo. The
detection of 3 T dwarfs in the limited spectroscopic follow-up of targets thus
far, and perhaps 1-2 strong candidates in photometric data (Greene et al.,
2023), is fully consistent with these simulations. Only $\sim$2 M dwarfs and
$\sim$1–2 L dwarfs are expected in the Abell 2744 field, in 1:1 and 2:1 ratios
of thin:thick disk members; and a negligible number of Y dwarfs due to their
intrinsic faintness. There are also few ($\sim$0-1) halo ultracool dwarfs
expected in this field despite its depth, although this prediction is more
sensitive to the poorly-constrained atmospheric and evolutionary properties of
metal-poor brown dwarfs. Hence, while the brown dwarf sample in the narrow
Abell 2744 field may be small, it is a mixture of different Galactic
populations that is distinct from the local Solar Neighborhood.

Our simulations allow us to assess the individual population memberships of
the three UNCOVER T dwarfs based on their estimated distances. Figure 3 shows
how the relative fractions of thin disk, thick disk, and halo T dwarfs varies
as a function of distance along the Abell 2744 line of sight. Beyond 1.4 kpc,
thick disk T dwarfs start to outnumber thin disk T dwarfs; beyond 6 kpc, halo
T dwarfs start to outnumber thick disk T dwarfs. The estimated distances of
the UNCOVER-BDs span these thresholds, with UNCOVER-BD-1 in particular having
76% and 24% probabilities of being a thick disk or halo member, respectively.
Our small spectral sample is therefore representative of brown dwarfs in the
Milky Way’s three main populations.

##  5 Discussion of Individual Sources

###  5.1 The Distant T subdwarf UNCOVER-BD-1

The subsolar metallicity features in the spectrum of UNCOVER-BD-1 are fully
consistent with this source’s probable association with the thick disk or
halo. The atmosphere parameters inferred for this source closely match those
of CWISE J1810-1010, a nearby T subdwarf identified in the citizen science
Backyard Worlds: Planet 9 program (Schneider et al., 2020; Kuchner et al.,
2017). We compare the spectrum of these two sources in Figure 4. The T
subdwarf more accurately reproduces the widened 1.1 $\mu$m absorption feature
and relatively featureless $H$-band peak characteristic of metal-poor brown
dwarfs, but shows excess flux on the blue wings of the $H$\- and $K$-band
peaks. Of course, it is unlikely given the few T subdwarfs currently known
that a perfect match would have occurred (Zhang et al., 2019a; Meisner et al.,
2021). Nevertheless, the similarities combined with both model fitting and
population analysis supports the interpretation of this source as a metal-poor
brown dwarf. We note that parallax measurement of CWISE J1810-1010 by Lodieu
et al. (2022) has shown this source to be considerably cooler ($\sim$800 K)
and closer (9 pc) then initial estimates, and the same may be true of our
estimates of UNCOVER-BD-1.

Figure 4: Comparison of the spectra of UNCOVER-BD-1 (black line) and the
extreme T subdwarf CWISE J1810-1010 (magenta line; data from Schneider et al.
2020). The latter is scaled to minimize the $\chi^{2}_{r}$ residuals between
the data. Note that breaks in the CWISE J1810-1010 spectrum represent regions
of strong telluric absorption in ground-based spectroscopy.

###  5.2 Phosphine in the Spectrum of UNCOVER-BD-3?

Our coldest source, UNCOVER-BD-3/GLASS-BD-1/Nonino’s Dwarf is notable for the
significant structure present in its 4–4.5 $\mu$m peak (Figure 5). This region
is shaped by multiple molecular species, including H2O, CH4, NH3, CO2, CO,
PH3, and H2S, with relative abundances influenced by non-equilibrium chemistry
and metallicity, (Griffith & Yelle, 1999; Noll et al., 2000; Saumon et al.,
2006; Visscher et al., 2006; Yamamura et al., 2010; Leggett et al., 2015;
Miles et al., 2020; Beiler et al., 2023). We compare the spectrum of UNCOVER-
BD-3 in this region to the JWST/NIRSpec spectrum of the Teff $\approx$ 450 K
Y0 dwarf WISE J035934.06-540154.6 (hereafter W0359-54; Kirkpatrick et al.
2012; Leggett et al. 2016; Beiler et al. 2023). Both sources show absorption
from the red wing of the 3.3 $\micron$ CH4 $\nu_{3}$ band, the 4.5 $\micron$
bandhead of CO (enhanced by mixing), and H2O extending toward and beyond 5
$\micron$. However, while W0359-54 shows a distinct minimum over 4.2–4.4
$\micron$, a feature that has been associated with CO2 absorption (Yamamura et
al., 2010; Sorahana & Yamamura, 2012; Beiler et al., 2023), this depression is
broader and lacks the sharp CO2 bandhead in the spectrum of for UNCOVER-BD-3.

We investigated this distinction by comparing absorption coefficients
$\alpha_{i}=n_{i}\sigma_{i}$ for the seven species $i$ listed above, using
cross-sections $\sigma_{i}$ at T = 600 K provided by the EXOMOL cross-section
server333See https://www.exomol.com/data/data-types/xsec. Cross-sections are
based on molecular line lists from Hill et al. (2013); Li et al. (2015);
Yurchenko et al. (2011); Polyansky et al. (2018); Azzam et al. (2016); Sousa-
Silva et al. (2015) and Yurchenko et al. (2020). (Hill et al., 2013; Tennyson
& Yurchenko, 2012). Relative abundances $n_{i}/n$ at T = 600 K and P = 10 bar
($n$ = 1.2$\times$1020 cm-3) are based on non-equilibrium chemistry for H2O,
CH4, CO, NH3 and PH3 (Saumon et al., 2006; Visscher et al., 2006), and
equilibrium chemistry for H2S (Visscher et al., 2006). For CO2, we adopted the
conjecture of Yamamura et al. (2010) that NCO2/NCO $\approx$ 10-3, and thus
exceeds equilibrium abundances by several orders of magnitude (Lodders &
Fegley, 2002). Yamamura et al. (2010) notes that this CO2 enrichment even
exceeds expectations for disequilibrium mixing abundances, but is necessary to
explain the distinct bandhead seen in solar-metallicity T and Y dwarf spectra
at 3–5 $\micron$. The mechanism for this enrichment has yet to be determined.
Indeed, Figure 5 shows that even with these assumptions, it is PH3, not CO2
that should be the primary absorber around 4.2 $\micron$. Introducting into
this analysis reduced metal abundances in the atmosphere of UNCOVER-BD-3,
suggested by our model fits and modest probability of thick-disk membership
(25%), would further reduce the abundance of CO2 relative to single-metal
species such as H2O or PH3, just as TiO is reduced relative to CaH in the
optical spectra of M-type subdwarfs (Mould & McElroy, 1978; Gizis, 1997).
While the blue wing of the 4.2 $\micron$ feature in UNCOVER-BD-3 could be
caused by a relative strengthening of H2O over CH4, we would expect to see
more absorption in the 4.8–5.0 $\micron$ region as well, but instead see a
slight rise in flux. Neither NH3 nor H2S show opacity trends that could
explain the structure of this feature.

We therefore claim that the 4.2 $\micron$ feature in UNCOVER-BD-3 is PH3, not
CO2, and its presence may serve as an indicator of subsolar metallicity among
the coldest brown dwarfs. The presence of multiple overlapping molecular
species in this region, whose abundances are sensitive to mixing and
metallicity, highlights its importance in atmospheric chemical studies of cold
brown dwarfs and warm exoplanets (Miles et al., 2023).

Figure 5: (Top panel) Absorption coefficients for H2O, CH4, NH3, CO2, CO, PH3,
and H2S in the 3.5–5 $\micron$ region based on EXOMOL cross-sections (Hill et
al., 2013; Tennyson & Yurchenko, 2012) and relative abundances based on
equilibrium and non-equilibrium chemistry (log values of relative abundance
for each species are listed in the legend; see text for details). The
strongest absorber at a given wavelength is toward the bottom of this plot.
Absorption coefficents for CO2 (in green) and PH3 (in brown) are highlighted.
(Bottom panel) Normalized spectra of UNCOVER-BD-3 (black line ) and the Y0
dwarf W0359-54 (magenta line) in the 3.5–5 $\micron$ region, with the former
offset for comparison (dashed lines). We identify the dominant absorbing
species across this band based on the top panel, including PH3 for UNCOVER-
BD-3 and CO2 for W0359-54.

##  6 Summary

We report JWST/NIRSpec spectroscopy of three distant T dwarfs identified in
the Abell 2744 lensing field by the UNCOVER JWST Legacy Survey. All three
objects display the characteristic signatures of low temperature brown dwarf
atmospheres, and are all well-matched in the 1–2.5 $\micron$ region to T1, T6,
and T8–T9 standards. Spectral model fits confirm these identifications, and
further indicate that UNCOVER-BD-1 and perhaps UNCOVER-BD-3 are metal-poor.
These fits provide robust estimates of the distances of these sources that
place them between 0.9–4.5 kpc, making them the most distant T dwarfs with
confirming spectroscopy to date. UNCOVER-BD-1 and UNCOVER-BD-2 in particular
are sufficiently distant to have a high probability of membership in the thick
disk. Similarities in the spectra of UNCOVER-BD-1 and the T subdwarf CWISE
J1810-1010 supports the metal-poor nature of both sources. For UNCOVER-BD-3,
the structure of the 4.2 $\micron$ peak indicates the signature of PH3
absorption rather than CO2 as seen in other T and Y dwarfs, and may reflect
metallicity effects on molecular chemistry in cold brown dwarf atmospheres.
Our population simulations indicate that the Abell 2744 field should contain
roughly 1–2 L-type and 5 T-type brown dwarfs, with one-third of these being
members of the Galactic thick disk. While modest, this reflects the yield from
a very small field of view (only seven NIRSpec pointings in this study), and
further investigations of an assemble of deep JWST pointings will uncover a
larger and statistically useful sample for characterizing the oldest, metal-
poor brown dwarfs in the Milky Way (Nonino et al., 2023; Hainline et al.,
2023; Holwerda et al., 2023).

We dedicate this paper to the memory of Dr. Mario Nonino, discoverer of the
first brown dwarf discovery made with JWST, GLASS-BD-1/UNCOVER-BD-3/Nonino’s
Dwarf, who passed way suddenly during the completion of this paper. Dr. Nonino
will be remembered as a friendly and collaborative colleague. This work is
based in part on observations made with the NASA/ESA _Hubble Space Telescope_
(HST) and the NASA/ESA/CSA _James Webb Space Telescope_ (JWST). Some of the
data presented in this paper were obtained from the Mikulski Archive for Space
Telescopes at the Space Telescope Science Institute. The specific observations
analyzed can be accessed via 10.17909/8k5c-xr27 (catalog 10.17909/8k5c-xr27).
These observations are associated with JWST programs JWST-GO-2561, JWST-
ERS-1324, and JWST-DD-2756; and with HST programs HST-GO-11689, HST-GO-13386,
HST-GO/DD-13495, HST-GO-13389, HST-GO-15117, and HST-GO/DD-17231. Support for
program JWST-GO-2561 was provided by NASA through a grant from the Space
Telescope Science Institute, which is operated by the Associations of
Universities for Research in Astronomy, Incorporated, under NASA contract NAS
5-03127. A.J.B. and R.G. acknowledge support from JWST program GO-2559 and
NASA/ADAP grant 21-ADAP21-0187. J.E.G. and A.D.G acknowledge support from
NSF/AAG grant # 1007094. J.E.G. also acknowledges support from NSF/AAG grant #
1007052. A.Z. acknowledges support by Grant No. 2020750 from the United
States-Israel Binational Science Foundation (BSF) and Grant No. 2109066 from
the United States National Science Foundation (NSF), and by the Ministry of
Science & Technology of Israel. The Cosmic Dawn Center is funded by the Danish
National Research Foundation (DNRF) under grant #140. This work has received
funding from the Swiss State Secretariat for Education, Research and
Innovation (SERI) under contract number MB22.00072, as well as from the Swiss
National Science Foundation (SNSF) through project grant 200020 207349. R.B.
acknowledges support from the Research Corporation for Scientific Advancement
(RCSA) Cottrell Scholar Award ID No: 27587 and from the National Science
Foundation NSF-CAREER grant # 2144314. P.D. acknowledges support from the
Dutch Research Council (NWO) through the award of the VIDI Grant
016.VIDI.189.162 (“ODIN”) and the European Commission’s and University of
Groningen’s CO-FUND Rosalind Franklin program. R.P.N. acknowledges funding
from JWST programs GO-1933 and GO-2279. Support for this work was provided by
NASA through the NASA Hubble Fellowship grant HST-HF2-51515.001-A awarded by
the Space Telescope Science Institute, which is operated by the Association of
Universities for Research in Astronomy, Incorporated, under NASA contract
NAS5-26555. The research of C.C.W. is supported by NOIRLab, which is managed
by the Association of Universities for Research in Astronomy (AURA) under a
cooperative agreement with the National Science Foundation. R.P. and D.M.
acknowledge support from JWST program GO-2561.

##  Appendix A Ultracool Dwarf Galactic Population Simulations

Here we describe in detail the ultracool dwarf population simulations
discussed in Section 4. These simulations are based on Monte Carlo approaches
previously presented in Burgasser (2004, 2007); Bardalez Gagliuffi et al.
(2019); Hsu et al. (2021); and Aganze et al. (2022), with additional
modifications described here to account for the spatial and spectral
properties of ultracool dwarfs in different Galactic populations. See Reid et
al. (1999); Day-Jones et al. (2013); Burningham et al. (2013); Ryan & Reid
(2016); van Vledder et al. (2016); Best et al. (2020); Kirkpatrick et al.
(2021); and Ryan et al. (2022) for other approaches.

We modeled our ultracool dwarf sample as three populations representing the
thin disk, the thick disk, and the halo, the latter treated as a single
population (cf. Carollo et al. 2008). We generated a set of 5$\times$105
masses between 0.01-0.1 M⊙ distributed as a power-law mass function with
$dN/dM\propto M^{-0.6}$ (Kirkpatrick et al., 2021) that were common to the
three populations; and three uniform distributions of ages, one spanning 0.5-8
Gyr for the thin disk, and two spanning 8–10 Gyr for the thick disk and halo.
These samples were then evolved to present-day temperatures ($T_{eff}$),
surface gravities ($\log{g}$), and luminosities using the Sonora-Bobcat models
(Marley et al., 2021), applying the [M/H] = 0 models for the thin disk and
[M/H] = $-$0.5 models for the thick disk and halo.444A comprehensive set of
lower metallicity evolutionary models for cold brown dwarfs using up-to-date
opacities are not yet publicly available; see Gerasimov et al. (2020).

These physical parameters were mapped to observational parameters of spectral
type and absolute magnitude using a combination of empirical relations and
theoretical atmosphere models. The empirical relations are based on a sample
of local, largely solar-metallicity ultracool dwarfs, and are most appropriate
for thin disk objects; see Zhang et al. (2019b) and Gonzales et al. (2021) for
relations appropriate for metal-poor M- and L-type ultracool dwarfs.
Temperatures were first mapped to spectral types from M6 to Y2 using an
updated calibration from555E. Mamajek 2022; see
https://www.pas.rochester.edu/~emamajek/EEM_dwarf_UBVIJHK_colors_Teff.txt.
Pecaut & Mamajek (2013). We then constructed spectral type/absolute magnitude
relations for six wide-field JWST/NIRCam filters, F115W, F150W, F200W, F277W,
F356W, and F444W, anchoring to existing spectral type/absolute magnitude
relations in proximate filters. For spectral types M6 to T7, we used the
spectral type/absolute magnitude relations of Dupuy & Liu (2012) for filters
MKO J, MKO H, MKO K, MKO LP, IRAC [3.6], and IRAC [4.5]. For spectral types T8
to Y2, we used the spectral type/absolute magnitude relations of Kirkpatrick
et al. (2021) for filters MKO J, 2MASS H, IRAC [3.6], and IRAC [4.5]. Color
terms between these filters and JWST/NIRCam filters were computed using
atmosphere models of the appropriate temperature and metallicity, and for
surface gravities 4.0 $\leq$ $\log~{}g$ $\leq$ 5.5 (in cm/s2). For Teff $>$
2400 K, we used the BT-Settl models (Allard et al., 2012); for Teff $\leq$
2400 K we used the Sonora-Bobcat models (Marley et al., 2021). Figure 6
displays the color terms between the empirical and JWST/NIRCam filters, and
Figure 7 displays the resulting JWST/NIRCam absolute magnitude/spectral type
relations, tabulated in Table 2. Uncertainties on these values vary as a
function of spectral type, and a conservative estimate of 0.5 mag is
appropriate for most of the spectral type range shown. Comparable relations
for a broader set of JWST/NIRCam filters can be found in Sanghi et al. (2023).

Figure 6: Filter color correction terms between JWST/NIRCam filters and
reference filters used for empirical relations as a function of Teff.
Corrections for a variety of surface gravities and for solar-scaled
metallicities [M/H] = 0 and -0.5 are shown by the various lines. Note that the
models used to compute these corrections changes from BT-Settl (Allard et al.,
2012) to Sonora-Bobcat (Marley et al., 2021) at Teff = 2400 K, and the
reference filters and absolute magnitude relations used change at spectral
type T8 (Teff = 700 K; see Table 2).  Figure 7: Absolute magnitudes in six
JWST/NIRCam filters as a function of spectral type adopted in this study. The
solid lines delineate the absolute magnitudes in the NIRCam filters in AB
magnitudes, while the dashed lines indicate the original absolute magnitudes
in the matched reference filter (indicated in the y-axis label and Table 2).
Note that the reference filter and empirical relation used changes at spectral
type T7/T8. The increasing divergence between reference filter and NIRCam
filter relations toward longer wavelengths largely reflects the divergence
between the Vega and AB magnitude systems.

Distances for each source were assigned by drawing from projected stellar
density distributions along the Abell 2744 line of sight using the Milky Way
structure parameters of Jurić et al. (2008). The maximum distances for these
distributions were set by the absolute F444W magnitude of each source and an
assumed limiting magnitude of 30 AB. For the M dwarfs in the simulation,
maximum distances can exceed 30 kpc; for the T dwarfs, maximum distances range
over 3–10 kpc. We ignored both unresolved multiplicity and statistical scatter
for these limits, factors that were considered insignificant given the small
size of our search area and final sample. Each source was assigned a distance
according to the line-of-sight density distribution for its population
truncated at this maximum distance, and then assigned a corresponding apparent
magnitude.

To construct a composite population, we randomly drew sources with replacement
from our thin disk, thick disk, and halo samples, ensuring that the relative
numbers of sources within 500 pc matched the local fractions in the Jurić et
al. (2008) model: thick disk/thin disk = 12% and halo/thin disk = 0.5%. We
further required that apparent magnitudes in at least three NIRCam filters
were brighter than the 30 AB limit to ensure multi-band detection. Finally, we
computed a scaling factor666The scaling factor $\frac{N_{\rm true}}{N_{\rm
sim}}=\frac{\rho_{obs}}{N_{\rm sim}(<d)}\frac{\Omega}{3}d^{3}$ assumes a
uniform spatial volume within our scaling distance $d$ = 100 pc; see
Kirkpatrick et al. (2019) and Best et al. (2020) for discussion on the spatial
isotropy of ultracool dwarfs within 20 pc of the Sun. to correct the
cumulative number of L1–T8 dwarfs within $d$ = 100 pc in our simulation to the
expected number of these sources in the same volume based on the measured
local space density $\rho_{obs}$ = (1.09$\pm$0.06)$\times$10-3 pc-3
(Kirkpatrick et al., 2021) and the imaged area of $\Omega$ = 45 arcmin2.

Table 2: JWST/NIRCam Absolute Magnitude Relations

SpT | Teff (K)aaTemperatures estimated from the spectral type/Teff relation of Pecaut & Mamajek (2013). | [M/H] | Model | F115W | F150W | F200W | F277W | F356W | F444W
---|---|---|---|---|---|---|---|---|---
Base magnitude M6-T7bbBased on the Dupuy & Liu (2012) absolute magnitude/spectral type relations. | MKO J | MKO H | MKO K | MKO LP | IRAC [3.6] | IRAC [4.5]
M6 | 2850 | 0 | BT | 10.918 | 10.769 | 10.985 | 11.985 | 11.911 | 12.328
M6 | 2850 | -0.5 | BT | 10.910 | 10.779 | 11.009 | 12.055 | 11.910 | 12.329
M7 | 2650 | 0 | BT | 11.742 | 11.585 | 11.631 | 12.354 | 12.085 | 12.467
M7 | 2650 | -0.5 | BT | 11.727 | 11.592 | 11.645 | 12.397 | 12.084 | 12.473
M8 | 2500 | 0 | BT | 12.168 | 12.012 | 12.022 | 12.664 | 12.285 | 12.651
M8 | 2500 | -0.5 | BT | 12.147 | 12.023 | 12.040 | 12.726 | 12.284 | 12.656
M9 | 2400 | 0 | SON | 12.461 | 12.205 | 12.254 | 13.044 | 12.502 | 12.843
M9 | 2400 | -0.5 | SON | 12.474 | 12.147 | 12.149 | 13.030 | 12.506 | 12.851
L0 | 2250 | 0 | SON | 12.706 | 12.420 | 12.458 | 13.404 | 12.722 | 13.052
L0 | 2250 | -0.5 | SON | 12.717 | 12.342 | 12.325 | 13.362 | 12.725 | 13.068
L1 | 2100 | 0 | SON | 12.990 | 12.675 | 12.665 | 13.732 | 12.939 | 13.270
L1 | 2100 | -0.5 | SON | 12.998 | 12.577 | 12.506 | 13.643 | 12.936 | 13.297
L2 | 1960 | 0 | SON | 13.338 | 12.999 | 12.892 | 13.996 | 13.141 | 13.499
L2 | 1960 | -0.5 | SON | 13.344 | 12.882 | 12.719 | 13.846 | 13.130 | 13.534
L3 | 1830 | 0 | SON | 13.748 | 13.384 | 13.147 | 14.198 | 13.326 | 13.727
L3 | 1830 | -0.5 | SON | 13.754 | 13.255 | 12.974 | 14.022 | 13.309 | 13.770
L4 | 1700 | 0 | SON | 14.201 | 13.806 | 13.421 | 14.353 | 13.494 | 13.954
L4 | 1700 | -0.5 | SON | 14.204 | 13.672 | 13.268 | 14.214 | 13.479 | 14.002
L5 | 1590 | 0 | SON | 14.660 | 14.224 | 13.716 | 14.496 | 13.656 | 14.175
L5 | 1590 | -0.5 | SON | 14.659 | 14.097 | 13.600 | 14.445 | 13.648 | 14.220
L6 | 1490 | 0 | SON | 15.090 | 14.606 | 14.026 | 14.659 | 13.812 | 14.382
L6 | 1490 | -0.5 | SON | 15.081 | 14.494 | 13.948 | 14.706 | 13.813 | 14.419
L7 | 1410 | 0 | SON | 15.451 | 14.928 | 14.339 | 14.855 | 13.966 | 14.568
L7 | 1410 | -0.5 | SON | 15.434 | 14.833 | 14.285 | 14.962 | 13.977 | 14.596
L8 | 1350 | 0 | SON | 15.714 | 15.176 | 14.637 | 15.066 | 14.124 | 14.732
L8 | 1350 | -0.5 | SON | 15.689 | 15.092 | 14.590 | 15.198 | 14.139 | 14.751
L9 | 1300 | 0 | SON | 15.865 | 15.342 | 14.901 | 15.273 | 14.286 | 14.880
L9 | 1300 | -0.5 | SON | 15.834 | 15.267 | 14.857 | 15.423 | 14.305 | 14.892
T0 | 1260 | 0 | SON | 15.908 | 15.437 | 15.134 | 15.493 | 14.461 | 15.014
T0 | 1260 | -0.5 | SON | 15.871 | 15.369 | 15.085 | 15.647 | 14.482 | 15.022
T1 | 1230 | 0 | SON | 15.861 | 15.485 | 15.341 | 15.712 | 14.655 | 15.142
T1 | 1230 | -0.5 | SON | 15.821 | 15.421 | 15.289 | 15.870 | 14.678 | 15.147
T2 | 1200 | 0 | SON | 15.765 | 15.522 | 15.545 | 15.958 | 14.878 | 15.273
T2 | 1200 | -0.5 | SON | 15.720 | 15.463 | 15.489 | 16.119 | 14.903 | 15.276
T3 | 1160 | 0 | SON | 15.673 | 15.603 | 15.780 | 16.269 | 15.142 | 15.417
T3 | 1160 | -0.5 | SON | 15.623 | 15.548 | 15.715 | 16.433 | 15.167 | 15.415
T4 | 1120 | 0 | SON | 15.650 | 15.783 | 16.084 | 16.630 | 15.459 | 15.582
T4 | 1120 | -0.5 | SON | 15.596 | 15.732 | 16.010 | 16.796 | 15.486 | 15.577
T5 | 1050 | 0 | SON | 15.773 | 16.122 | 16.507 | 17.131 | 15.842 | 15.782
T5 | 1050 | -0.5 | SON | 15.713 | 16.077 | 16.416 | 17.304 | 15.870 | 15.772
T6 | 960 | 0 | SON | 16.116 | 16.672 | 17.099 | 17.780 | 16.308 | 16.028
T6 | 960 | -0.5 | SON | 16.054 | 16.634 | 16.982 | 17.987 | 16.337 | 16.011
T7 | 840 | 0 | SON | 16.754 | 17.472 | 17.907 | 18.678 | 16.874 | 16.335
T7 | 840 | -0.5 | SON | 16.702 | 17.438 | 17.747 | 18.891 | 16.902 | 16.314
Base magnitude T8-Y2ccBased on the Kirkpatrick et al. (2021) absolute magnitude/spectral type relations. | MKO J | 2MASS H | 2MASS H | IRAC [3.6] | IRAC [3.6] | IRAC [4.5]
T8 | 700 | 0 | SON | 17.393 | 19.593 | 20.479 | 20.010 | 17.909 | 16.907
T8 | 700 | -0.5 | SON | 17.365 | 19.538 | 21.128 | 20.329 | 17.940 | 16.881
T9 | 530 | 0 | SON | 18.808 | 21.234 | 22.785 | 21.909 | 18.772 | 17.483
T9 | 530 | -0.5 | SON | 18.851 | 21.193 | 23.583 | 22.095 | 18.813 | 17.449
Y0 | 420 | 0 | SON | 20.733 | 23.561 | 26.062 | 23.767 | 19.687 | 18.159
Y0 | 420 | -0.5 | SON | 20.899 | 23.533 | 26.853 | 23.787 | 19.738 | 18.121
Y1 | 350 | 0 | SON | 23.362 | 27.059 | 30.632 | 25.362 | 20.476 | 18.916
Y1 | 350 | -0.5 | SON | 23.519 | 27.048 | 31.020 | 25.201 | 20.541 | 18.876
Y2 | 250 | 0 | SON | 26.810 | 32.485 | 38.047 | 26.911 | 20.737 | 19.743
Y2 | 250 | -0.5 | SON | 26.356 | 32.489 | 37.352 | 26.796 | 20.868 | 19.699

Note. — Model references: [BT] BT-Settl models (Allard et al., 2012); [SON]
Sonora-Bobcat models (Marley et al., 2021).

## References

  * Aganze et al. (2022) Aganze, C., Burgasser, A. J., Malkan, M., et al. 2022, ApJ, 934, 73, doi: 10.3847/1538-4357/ac7053
  * Allard et al. (2012) Allard, F., Homeier, D., & Freytag, B. 2012, Philosophical Transactions of the Royal Society A, 370, 2765, doi: 10.1098/rsta.2011.0269
  * Astropy Collaboration et al. (2013) Astropy Collaboration, Robitaille, T. P., Tollerud, E. J., et al. 2013, A&A, 558, A33, doi: 10.1051/0004-6361/201322068
  * Astropy Collaboration et al. (2018) Astropy Collaboration, Price-Whelan, A. M., Sipőcz, B. M., et al. 2018, AJ, 156, 123, doi: 10.3847/1538-3881/aabc4f
  * Astropy Collaboration et al. (2022) Astropy Collaboration, Price-Whelan, A. M., Lim, P. L., et al. 2022, ApJ, 935, 167, doi: 10.3847/1538-4357/ac7c74
  * Atek et al. (2023) Atek, H., Labbé, I., Furtak, L. J., et al. 2023, arXiv e-prints, arXiv:2308.08540, doi: 10.48550/arXiv.2308.08540
  * Azzam et al. (2016) Azzam, A. A. A., Tennyson, J., Yurchenko, S. N., & Naumenko, O. V. 2016, MNRAS, 460, 4063, doi: 10.1093/mnras/stw1133
  * Bardalez Gagliuffi et al. (2019) Bardalez Gagliuffi, D. C., Burgasser, A. J., Schmidt, S. J., et al. 2019, ApJ, 883, 205, doi: 10.3847/1538-4357/ab253d
  * Beiler et al. (2023) Beiler, S. A., Cushing, M. C., Kirkpatrick, J. D., et al. 2023, ApJ, 951, L48, doi: 10.3847/2041-8213/ace32c
  * Best et al. (2020) Best, W. M. J., Liu, M. C., Magnier, E. A., & Dupuy, T. J. 2020, AJ, 159, 257, doi: 10.3847/1538-3881/ab84f4
  * Bezanson et al. (2022) Bezanson, R., Labbe, I., Whitaker, K. E., et al. 2022, arXiv e-prints, arXiv:2212.04026, doi: 10.48550/arXiv.2212.04026
  * Brammer (2022) Brammer, G. 2022, msaexp: NIRSpec analyis tools, 0.3.4, Zenodo, Zenodo, doi: 10.5281/zenodo.7313329
  * Burgasser (2004) Burgasser, A. J. 2004, ApJS, 155, 191, doi: 10.1086/424386
  * Burgasser (2007) —. 2007, ApJ, 659, 655, doi: 10.1086/511027
  * Burgasser et al. (2006a) Burgasser, A. J., Burrows, A., & Kirkpatrick, J. D. 2006a, ApJ, 639, 1095, doi: 10.1086/499344
  * Burgasser et al. (2006b) Burgasser, A. J., Geballe, T. R., Leggett, S. K., Kirkpatrick, J. D., & Golimowski, D. A. 2006b, ApJ, 637, 1067, doi: 10.1086/498563
  * Burgasser et al. (2004) Burgasser, A. J., McElwain, M. W., Kirkpatrick, J. D., et al. 2004, AJ, 127, 2856, doi: 10.1086/383549
  * Burgasser & Splat Development Team (2017) Burgasser, A. J., & Splat Development Team. 2017, in Astronomical Society of India Conference Series, Vol. 14, Astronomical Society of India Conference Series, 7–12.  https://arxiv.org/abs/1707.00062
  * Burningham et al. (2013) Burningham, B., Cardoso, C. V., Smith, L., et al. 2013, MNRAS, 433, 457, doi: 10.1093/mnras/stt740
  * Bushouse et al. (2023) Bushouse, H., Eisenhamer, J., Dencheva, N., et al. 2023, JWST Calibration Pipeline, 1.11.3, Zenodo, doi: 10.5281/zenodo.8157276
  * Carollo et al. (2008) Carollo, D., Beers, T. C., Lee, Y. S., et al. 2008, Nature, 451, 216, doi: 10.1038/nature06542
  * Day-Jones et al. (2013) Day-Jones, A. C., Marocco, F., Pinfield, D. J., et al. 2013, MNRAS, 430, 1171, doi: 10.1093/mnras/sts685
  * Dupuy & Liu (2012) Dupuy, T. J., & Liu, M. C. 2012, ApJS, 201, 19, doi: 10.1088/0067-0049/201/2/19
  * Furtak et al. (2023) Furtak, L. J., Labbé, I., Zitrin, A., et al. 2023, arXiv e-prints, arXiv:2308.05735, doi: 10.48550/arXiv.2308.05735
  * Gerasimov et al. (2022) Gerasimov, R., Burgasser, A. J., Homeier, D., et al. 2022, ApJ, 930, 24, doi: 10.3847/1538-4357/ac61e5
  * Gerasimov et al. (2020) Gerasimov, R., Homeier, D., Burgasser, A., & Bedin, L. R. 2020, Research Notes of the American Astronomical Society, 4, 214, doi: 10.3847/2515-5172/abcf2c
  * Gizis (1997) Gizis, J. E. 1997, AJ, 113, 806, doi: 10.1086/118302
  * Glazebrook et al. (2023) Glazebrook, K., Nanayakkara, T., Jacobs, C., et al. 2023, ApJ, 947, L25, doi: 10.3847/2041-8213/acba8b
  * Gonzales et al. (2021) Gonzales, E. C., Burningham, B., Faherty, J. K., et al. 2021, ApJ, 923, 19, doi: 10.3847/1538-4357/ac294e
  * Goulding et al. (2023) Goulding, A. D., Greene, J. E., Setton, D. J., et al. 2023, arXiv e-prints, arXiv:2308.02750, doi: 10.48550/arXiv.2308.02750
  * Greene et al. (2023) Greene, J. E., Labbe, I., Goulding, A. D., et al. 2023, arXiv e-prints, arXiv:2309.05714, doi: 10.48550/arXiv.2309.05714
  * Griffith & Yelle (1999) Griffith, C. A., & Yelle, R. V. 1999, ApJ, 519, L85, doi: 10.1086/312103
  * Hainline et al. (2023) Hainline, K. N., Helton, J. M., Johnson, B. D., et al. 2023, arXiv e-prints, arXiv:2309.03250, doi: 10.48550/arXiv.2309.03250
  * Hayashi & Nakano (1963) Hayashi, C., & Nakano, T. 1963, Progress of Theoretical Physics, 30, 460, doi: 10.1143/PTP.30.460
  * Hill et al. (2013) Hill, C., Yurchenko, S. N., & Tennyson, J. 2013, Icarus, 226, 1673, doi: 10.1016/j.icarus.2012.07.028
  * Holwerda et al. (2023) Holwerda, B. W., Hsu, C.-C., Hathi, N., et al. 2023, arXiv e-prints, arXiv:2309.05835, doi: 10.48550/arXiv.2309.05835
  * Horne (1986) Horne, K. 1986, PASP, 98, 609, doi: 10.1086/131801
  * Hsu et al. (2021) Hsu, C.-C., Burgasser, A. J., Theissen, C. A., et al. 2021, ApJS, 257, 45, doi: 10.3847/1538-4365/ac1c7d
  * Jurić et al. (2008) Jurić, M., Ivezić, Ž., Brooks, A., et al. 2008, ApJ, 673, 864, doi: 10.1086/523619
  * Karalidi et al. (2021) Karalidi, T., Marley, M., Fortney, J. J., et al. 2021, ApJ, 923, 269, doi: 10.3847/1538-4357/ac3140
  * Kirkpatrick et al. (2012) Kirkpatrick, J. D., Gelino, C. R., Cushing, M. C., et al. 2012, ApJ, 753, 156, doi: 10.1088/0004-637X/753/2/156
  * Kirkpatrick et al. (2019) Kirkpatrick, J. D., Martin, E. C., Smart, R. L., et al. 2019, ApJS, 240, 19, doi: 10.3847/1538-4365/aaf6af
  * Kirkpatrick et al. (2021) Kirkpatrick, J. D., Gelino, C. R., Faherty, J. K., et al. 2021, ApJS, 253, 7, doi: 10.3847/1538-4365/abd107
  * Kuchner et al. (2017) Kuchner, M. J., Faherty, J. K., Schneider, A. C., et al. 2017, ApJ, 841, L19, doi: 10.3847/2041-8213/aa7200
  * Kumar (1962) Kumar, S. S. 1962, AJ, 67, 579, doi: 10.1086/108658
  * Kumar (1963) —. 1963, ApJ, 137, 1121, doi: 10.1086/147589
  * Labbe et al. (2023) Labbe, I., Greene, J. E., Bezanson, R., et al. 2023, arXiv e-prints, arXiv:2306.07320, doi: 10.48550/arXiv.2306.07320
  * Langeroodi & Hjorth (2023) Langeroodi, D., & Hjorth, J. 2023, arXiv e-prints, arXiv:2308.10900, doi: 10.48550/arXiv.2308.10900
  * Leggett et al. (2015) Leggett, S. K., Morley, C. V., Marley, M. S., & Saumon, D. 2015, ApJ, 799, 37, doi: 10.1088/0004-637X/799/1/37
  * Leggett et al. (2016) Leggett, S. K., Tremblin, P., Saumon, D., et al. 2016, ApJ, 824, 2, doi: 10.3847/0004-637X/824/1/2
  * Li et al. (2015) Li, G., Gordon, I. E., Rothman, L. S., et al. 2015, ApJS, 216, 15, doi: 10.1088/0067-0049/216/1/15
  * Linsky (1969) Linsky, J. L. 1969, ApJ, 156, 989, doi: 10.1086/150030
  * Lodders & Fegley (2002) Lodders, K., & Fegley, B. 2002, Icarus, 155, 393, doi: 10.1006/icar.2001.6740
  * Lodieu et al. (2022) Lodieu, N., Zapatero Osorio, M. R., Martín, E. L., Rebolo López, R., & Gauza, B. 2022, A&A, 663, A84, doi: 10.1051/0004-6361/202243516
  * Marley et al. (2021) Marley, M. S., Saumon, D., Visscher, C., et al. 2021, ApJ, 920, 85, doi: 10.3847/1538-4357/ac141d
  * Meisner et al. (2021) Meisner, A. M., Schneider, A. C., Burgasser, A. J., et al. 2021, ApJ, 915, 120, doi: 10.3847/1538-4357/ac013c
  * Miles et al. (2020) Miles, B. E., Skemer, A. J. I., Morley, C. V., et al. 2020, AJ, 160, 63, doi: 10.3847/1538-3881/ab9114
  * Miles et al. (2023) Miles, B. E., Biller, B. A., Patapis, P., et al. 2023, ApJ, 946, L6, doi: 10.3847/2041-8213/acb04a
  * Mould & McElroy (1978) Mould, J. R., & McElroy, D. B. 1978, ApJ, 220, 935, doi: 10.1086/155983
  * Noll et al. (2000) Noll, K. S., Geballe, T. R., Leggett, S. K., & Marley, M. S. 2000, ApJ, 541, L75, doi: 10.1086/312906
  * Nonino et al. (2023) Nonino, M., Glazebrook, K., Burgasser, A. J., et al. 2023, ApJ, 942, L29, doi: 10.3847/2041-8213/ac8e5f
  * Pecaut & Mamajek (2013) Pecaut, M. J., & Mamajek, E. E. 2013, ApJS, 208, 9, doi: 10.1088/0067-0049/208/1/9
  * Phillips et al. (2020) Phillips, M. W., Tremblin, P., Baraffe, I., et al. 2020, A&A, 637, A38, doi: 10.1051/0004-6361/201937381
  * Polyansky et al. (2018) Polyansky, O. L., Kyuberis, A. A., Zobov, N. F., et al. 2018, MNRAS, 480, 2597, doi: 10.1093/mnras/sty1877
  * Reid et al. (1999) Reid, I. N., Kirkpatrick, J. D., Liebert, J., et al. 1999, ApJ, 521, 613, doi: 10.1086/307589
  * Reylé et al. (2021) Reylé, C., Jardine, K., Fouqué, P., et al. 2021, A&A, 650, A201, doi: 10.1051/0004-6361/202140985
  * Ryan & Reid (2016) Ryan, R. E., J., & Reid, I. N. 2016, AJ, 151, 92, doi: 10.3847/0004-6256/151/4/92
  * Ryan et al. (2022) Ryan, R. E., Thorman, P., Aganze, C., et al. 2022, ApJ, 932, 96, doi: 10.3847/1538-4357/ac6de5
  * Sanghi et al. (2023) Sanghi, A., Liu, M. C., Best, W. M., et al. 2023, Research Notes of the American Astronomical Society, 7, 194, doi: 10.3847/2515-5172/acf864
  * Saumon et al. (2006) Saumon, D., Marley, M. S., Cushing, M. C., et al. 2006, ApJ, 647, 552, doi: 10.1086/505419
  * Schneider et al. (2020) Schneider, A. C., Burgasser, A. J., Gerasimov, R., et al. 2020, ApJ, 898, 77, doi: 10.3847/1538-4357/ab9a40
  * Sorahana & Yamamura (2012) Sorahana, S., & Yamamura, I. 2012, ApJ, 760, 151, doi: 10.1088/0004-637X/760/2/151
  * Sousa-Silva et al. (2015) Sousa-Silva, C., Al-Refaie, A. F., Tennyson, J., & Yurchenko, S. N. 2015, MNRAS, 446, 2337, doi: 10.1093/mnras/stu2246
  * Stauffer et al. (1998) Stauffer, J. R., Schultz, G., & Kirkpatrick, J. D. 1998, ApJ, 499, L199+, doi: 10.1086/311379
  * Tennyson & Yurchenko (2012) Tennyson, J., & Yurchenko, S. N. 2012, MNRAS, 425, 21, doi: 10.1111/j.1365-2966.2012.21440.x
  * van Vledder et al. (2016) van Vledder, I., van der Vlugt, D., Holwerda, B. W., et al. 2016, MNRAS, 458, 425, doi: 10.1093/mnras/stw258
  * Visscher et al. (2006) Visscher, C., Lodders, K., & Fegley, B. J. 2006, ApJ, 648, 1181, doi: 10.1086/506245
  * Wang et al. (2023a) Wang, B., Fujimoto, S., Labbe, I., et al. 2023a, arXiv e-prints, arXiv:2308.03745, doi: 10.48550/arXiv.2308.03745
  * Wang et al. (2023b) Wang, P.-Y., Goto, T., Ho, S. C. C., et al. 2023b, MNRAS, 523, 4534, doi: 10.1093/mnras/stad1679
  * Weaver et al. (2023) Weaver, J. R., Cutler, S. E., Pan, R., et al. 2023, arXiv e-prints, arXiv:2301.02671, doi: 10.48550/arXiv.2301.02671
  * Yamamura et al. (2010) Yamamura, I., Tsuji, T., & Tanabé, T. 2010, ApJ, 722, 682, doi: 10.1088/0004-637X/722/1/682
  * Yurchenko et al. (2011) Yurchenko, S. N., Barber, R. J., & Tennyson, J. 2011, MNRAS, 413, 1828, doi: 10.1111/j.1365-2966.2011.18261.x
  * Yurchenko et al. (2020) Yurchenko, S. N., Mellor, T. M., Freedman, R. S., & Tennyson, J. 2020, MNRAS, 496, 5282, doi: 10.1093/mnras/staa1874
  * Zhang et al. (2019a) Zhang, S., Luo, A. L., Comte, G., et al. 2019a, ApJS, 240, 31, doi: 10.3847/1538-4365/aafb32
  * Zhang et al. (2019b) Zhang, Z. H., Burgasser, A. J., Gálvez-Ortiz, M. C., et al. 2019b, MNRAS, 486, 1260, doi: 10.1093/mnras/stz777