\begin{proposition}\label{prop: rank 3 excluded minor}
    If $M$ is a simple rank-$3$ matroid with a line $L$ and distinct points $e_1, e_2, e_3$ on $L$ each on at least two long lines, then $M$ is not a positroid.
    In particular, $M$ has a minor from Figure \ref{fig: rank-3 excluded minors}.
\end{proposition}

\begin{theorem} \label{thm: main}
For all integers $r, \ell \ge 1$, if $M$ is a simple rank-$r$ positroid with no $U_{2,\ell+2}$-minor, then $|M| \le \ell(r - 1) + 1$.
Moreover, if $r \ge 2$ then equality holds if and only if $M$ can be obtained by taking parallel connections of copies of $U_{2, \ell+1}$.
\end{theorem}

\begin{proposition}\label{prop: rank 4 excluded minor}
    If $M$ is a simple rank-$4$ matroid with a line $L$, distinct points $e_1, e_2, e_3$ on $L$, and distinct planes $P_1$, $P_2$, $P_3$ through $L$ so that for $i = 1,2,3$ the point $e_i$ is on at least two long lines in $P_i$, then $M$ is not a positroid.
    In particular, $M$ has a minor from Figure \ref{fig: rank-3 excluded minors} or Figure \ref{fig: rank-4 excluded minor}.
\end{proposition}

\label{sec: introduction}

Extremal matroid theory was formally introduced in 1993 with the following problem posed by Joseph Kung \cite{Kung-1993}.

\begin{problem} \label{prob: Kung problem}
Let $\cM$ be a class of matroids.
Determine the maximum number of elements of a simple rank-$r$ matroid in $\cM$, and characterize the matroids for which equality holds.
\end{problem}

This maximum does not exist for all classes of matroids.
For instance, there is no finite answer to Problem~\ref{prob: Kung problem} for the class of all matroids, or for any class containing all uniform matroids.
However, Kung \cite[Theorem 4.3]{Kung-1993} proved that every simple rank-$r$ matroid with no $U_{2, \ell+2}$-minor has at most $\frac{\ell^r - 1}{\ell - 1}$ elements.
If $\ell$ is a prime power, then this bound is tight and is achieved by the rank-$r$ projective geometry over the finite field of order $\ell$. It is thus natural to study Problem~\ref{prob: Kung problem} for classes of matroids without $U_{2,\ell}$-minors.

In this paper, we study the class of \textit{positroids} without $U_{2,\ell}$-minors. Positroids, which are matroids representable by a matrix over $\bR$ with all maximal subdeterminants nonnegative, occupy a central position at the intersection of combinatorics, algebraic geometry, and physics. They serve as combinatorial models for the totally nonnegative Grassmannian, which in turn underlies the construction of the amplituhedron in quantum field theory. The study of amplituhedra and related objects remains an active and influential area in theoretical physics \cite{AHT}. Through these connections, positroids emerge as unifying objects that integrate discrete combinatorial frameworks with geometric and physical theory. Combinatorially, they are remarkably rich: Postnikov~\cite{postnikov} showed that positroids are in bijection with decorated permutations, Grassmann necklaces, plabic graphs, and \reflectbox{L}-diagrams, providing multiple equivalent viewpoints.
Positroids also form a minor-closed class of matroids. It is thus natural to ask extremal questions about positroids and their minors. There has been work enumerating both positroids \cite{Williams05} and connected positroids \cite{ARW16}, but as far as the authors are aware, no other extremal results about positroids are known. Our main result tightly resolves Problem~\ref{prob: Kung problem} for the class of $U_{2,\ell+2}$-minor-free positroids.

This result lies at the intersection of three areas of research: extremal matroid theory, structural properties of positroids, and structural properties of matroids with no $M(K_4)$-minor.
We will explain how Theorem \ref{thm: main} continues lines of research in these three areas, beginning with extremal matroid theory.
The following beautiful result of Geelen and Nelson \cite[Theorem 1.2]{Geelen-Nelson-2010} generalizes the initial work of Kung and tightly solves Problem \ref{prob: Kung problem} for the class of matroids with no $U_{2,\ell+2}$-minor, for all values of $\ell$ and all sufficiently large values of $r$.

\begin{theorem}[Geelen, Nelson \cite{Geelen-Nelson-2010}] \label{thm: exlude a line}
Let $\ell\ge 2$ be an integer and let $q$ be the largest prime power at most $\ell$.
For $r$ sufficiently large, every simple rank-$r$ matroid with no $U_{2,\ell+2}$-minor has at most $\frac{q^r - 1}{q - 1}$ elements, with equality only for the rank-$r$ projective geometry over the order-$q$ finite field.
\end{theorem}

\begin{theorem}[Geelen, Nelson, Walsh \cite{Geelen-Nelson-Walsh-2024}] \label{thm: complex exclude a line}
Let $\ell\ge 2$ be an integer and let $r$ be a sufficiently large integer.
Then every simple rank-$r$ $\bC$-representable matroid with no $U_{2,\ell+2}$-minor has at most $\ell\binom{r}{2} + r$ elements, with equality only for the rank-$r$ Dowling geometry over the cyclic group of order $\ell$.
\end{theorem}

\begin{theorem} \label{thm: main result with proof}
For all integers $r, \ell \ge 1$, if $M$ is a simple rank-$r$ matroid with no minor from Figure \ref{fig: rank-3 excluded minors} or Figure \ref{fig: rank-4 excluded minor} and no $U_{2,\ell+2}$-minor, then $|M| \le \ell(r - 1) + 1$.
Moreover, equality holds if and only if $M$ can be obtained by taking parallel connections of copies of $U_{2, \ell+1}$.
\end{theorem}
\begin{proof}
First suppose that $\ell = 1$.
If $M$ is a simple rank-$r$ matroid with more than $r$ elements, then $r \ge 2$ and $M$ has a circuit with at least three elements.
But every circuit with at least three elements has a $U_{2,3}$-minor \cite[Example 3.1.9]{Oxley-2011}, a contradiction.
So $M$ is a rank-$r$ matroid with $r$ elements, and it follows that $M$ is the parallel connection of $r - 1$ copies of $U_{2,2}$, as desired.
So we may assume that $\ell \ge 2$.
We will proceed by induction on $r$.
The statement is clearly true when $r \le 2$ so we may assume that $r \ge 3$.
If $M$ has an element $e$ that is not on any long lines, then $M/e$ is simple, $|M/e| = |M| - 1$, and $r(M/e) = r - 1$, so by induction we have
\begin{align*}
 |M| = 1 + |M/e| \le 1 + \ell(r - 2) + 1 = \ell(r - 1) + 2 - \ell < \ell(r - 1) + 1,
\end{align*}
as desired.
So we may assume that every element of $M$ is on a long line.
Let $L$ be a long line of $M$.
We will now consider the connected components of $M/L$.
Let $\cX$ be the partition of $E(M) - L$ given by the ground sets of the connected components of $M/L$.
Let $X \in \cX$, and let $N = M|(L \cup X)$.
We will first bound the number of elements of $N \del L$.

\begin{claim} \label{claim: bound the size of each component}
 At most two elements in $L$ are on a long line of $N$ other than $L$, and $|N \del L| \le \ell(r(N) - 2)$.
\end{claim}
\begin{proof}
Suppose that distinct elements $e_1$, $e_2$, and $e_3$ in $L$ are on long lines $L_1$, $L_2$, and $L_3$, respectively, of $N$ other than $L$.
Let $K = \si(N/L)$.
By deleting elements from nontrivial parallel classes of $N/L$ we may assume that $K$ is a restriction of $N/L$.
Note that the elements of $K$ are in one-to-one correspondence with the planes of $N$ containing $L$.
For $i = 1,2,3$ let $p_i$ be the element of $K$ in the plane $\cl_N(L \cup L_i)$ of $N$.
Since $K$ is connected, it follows from \cite[Theorem 4.3.1]{Oxley-2011} that $K$ has a connected minor $K_1 = K \del D/C$ on ground set $\{p_1, p_2, p_3\}$.
We may assume that $C$ is independent in $K$ and therefore in $N/L$, which implies that $r_{N/C}(L) = r_N(L) = 2$.
Since $K_1$ is connected, it is isomorphic to $U_{1,3}$ or $U_{2,3}$.
Let $N_1$ be obtained from $N$ by deleting, for each element $d$ in $D$, the elements in $\cl_N(L \cup d) - L$, and then contracting $C$.
Then $K_1 = \si(N_1/L)$ and $r(N_1) = r(K_1) + 2$.
We will show that $N_1$ has a minor from Figure \ref{fig: rank-3 excluded minors} or \ref{fig: rank-4 excluded minor}.

\begin{claim}
 Some $x \in L$ is on more than one long line of $M$.
\end{claim}
\begin{proof}
Let $e \in L$ and let $M_1 = M/e$.
Let $P = L - e$, and note that $P$ is a nontrivial parallel class of $M_1$.
Also note that $\elem(M_1) = |M| - \ell = \ell(r(M_1) - 1) + 1$,
so by induction, $\si(M_1)$ is obtained by taking parallel connections of copies of $U_{2,\ell+1}$.
This implies that $P$ is on a long line $L_1$ of $M_1$.
Let $F$ be the plane $L \cup L_1$ of $M$.
Note that $|F| \ge |L| + |L_1| - 1 \ge 2(\ell + 1) - 1 = 2\ell + 1$.
If $F - L$ has a $3$-element independent set $I$, then $M|F$ has an $M(K_4)$-restriction (if each pair of points in $I$ spans a point on $L$) or a $U_{2, \ell+2}$-minor, a contradiction.
So $F - L$ has rank $2$.
If $\cl(F - L)$ does not intersect $L$, then for every $f \in F - L$ we have $(M|F)/f \cong U_{2,\ell+2}$, a contradiction.
So $\cl(F - L)$ intersects $L$ in some element $x$.
Since $|F - L| \ge \ell \ge 2$, it follows that $\cl(F - L)$ is a long line of $M$ that contains $x$ and is not equal to $L$, as desired.
\end{proof}

Let $r, \ell$ be integers with $r \ge 2$ and $\ell \ge 3$.
Let $W(r,\ell)$ be the matroid with a basis $B = \{b_1, b_2, \dots, b_r\}$ and $\lfloor \frac{\ell - 1}{2} \rfloor$ elements freely placed (see \cite[page 270]{Oxley-2011}) in the span of $\{b_i, b_{i+1}\}$ for all $i \in [r]$, reading indices modulo $r$.
Let $W(r, \ell)^+$ be obtained from $W(r, \ell)$ by freely placing one point in $\cl(\{b_1, b_2\})$.
Then $W(2, \ell) \cong U_{2,\lfloor \frac{\ell - 1}{2} \rfloor + 2}$, and $W(r, 3)$ and $W(r, 4)$ are both isomorphic to the rank-$r$ whirl $\mathcal W^r$ when $r \ge 3$.
Since the whirl $\mathcal W^r$ is $3$-connected (see \cite[Example 8.3.4]{Oxley-2011}) and $W(r,\ell)$ has $\mathcal W^r$ as a restriction, it follows from \cite[page 295, Exercise 5]{Oxley-2011} that $W(r,\ell)$ and $W(r, \ell)^+$ are $3$-connected.
And it is straightforward to check that the natural cyclic ordering of $E(W(r, \ell))$ (first $b_1$, then all elements in $\cl(\{b_1, b_2\}) - b_2$, then $b_2$, and so on) or $W(r,\ell)^+$ satisfies the condition of Theorem \ref{thm:Bonin's condition}, so $W(r, \ell)$ and $W(r, \ell)^+$ are positroids.
Finally, one can check that if $r \ge 4$ and $e \notin \{b_1, \dots, b_r\}$, then $\si(W(r, \ell)/e) \cong W(r, \ell)$, and that $\si(W(r, \ell)/b_i) \cong W(r,\ell) \del (\cl(\{b_{i-1},b_i, b_{i+1}) - \{b_{i-1}, b_{i+1}\})$, and from these observations one can prove that $W(r,\ell)$ is $U_{2,\ell+2}$-minor-free, and when $\ell$ is even, $W(r, \ell)^+$ is $U_{2, \ell+2}$-minor-free.
Noting that $|W(r, \ell)| = r + r\lfloor \frac{\ell-1}{2} \rfloor$, we make the following conjecture.

This result lies at the intersection of three areas of research: extremal matroid theory, structural properties of positroids, and structural properties of matroids with no $M(K_4)$-minor.
We will explain how Theorem \ref{thm: main} continues lines of research in these three areas, beginning with extremal matroid theory.
The following beautiful result of Geelen and Nelson \cite[Theorem 1.2]{Geelen-Nelson-2010} generalizes the initial work of Kung and tightly solves Problem \ref{prob: Kung problem} for the class of matroids with no $U_{2,\ell+2}$-minor, for all values of $\ell$ and all sufficiently large values of $r$.

\begin{theorem}[Geelen, Nelson \cite{Geelen-Nelson-2010}] \label{thm: exlude a line}
Let $\ell\ge 2$ be an integer and let $q$ be the largest prime power at most $\ell$.
For $r$ sufficiently large, every simple rank-$r$ matroid with no $U_{2,\ell+2}$-minor has at most $\frac{q^r - 1}{q - 1}$ elements, with equality only for the rank-$r$ projective geometry over the order-$q$ finite field.
\end{theorem}

Projective geometries are not representable over fields of characteristic zero (see \cite[page 205, Exercise 8]{Oxley-2011}), inviting the following question: for a field $\bF$ with characteristic zero, what is the maximum number of elements of a simple rank-$r$ $\bF$-representable matroid with no $U_{2, \ell+2}$-minor?
This was answered by Geelen, Nelson, and Walsh \cite[Theorem 1.0.5]{Geelen-Nelson-Walsh-2024} in the case that $\bF = \bC$.

\begin{theorem}[Geelen, Nelson, Walsh \cite{Geelen-Nelson-Walsh-2024}] \label{thm: complex exclude a line}
Let $\ell\ge 2$ be an integer and let $r$ be a sufficiently large integer.
Then every simple rank-$r$ $\bC$-representable matroid with no $U_{2,\ell+2}$-minor has at most $\ell\binom{r}{2} + r$ elements, with equality only for the rank-$r$ Dowling geometry over the cyclic group of order $\ell$.
\end{theorem}

When $\bF \in \{\bR, \mathbb Q\}$, there is a lower bound of $2\binom{r}{2} + r$ and an upper bound of $2 \binom{r}{2} + f(\ell)\cdot r$ with $f$ a function of $\ell$, due to Geelen, Nelson, and Walsh \cite[Theorem 1.0.6]{Geelen-Nelson-Walsh-2024}, but no exact solution to Problem~\ref{prob: Kung problem} is known for all $\ell$.
Theorem \ref{thm: main} can be seen as a further specialization of Theorems \ref{thm: exlude a line} and \ref{thm: complex exclude a line} to the subclass of $\bR$-representable matroids consisting of positroids.

Theorem \ref{thm: main} also continues a recent line of research that studies positroids from a structural perspective.
For example, Bonin \cite{Bonin-2024} studied matroid amalagams of positroids, Quail \cite{Quail-2024} studied positroids representable over the finite fields of order $3$ and of order $4$, Quail and Rombach \cite{Quail-Rombach-2025} studied graphic positroids, and Park \cite{Park-2023} studied excluded minors for positroids.
Our proof of Theorem \ref{thm: main} relies on two new structural properties (Propositions \ref{prop: rank 3 excluded minor} and \ref{prop: rank 4 excluded minor}) for low-rank positroids.

Finally, \Cref{thm: main} contributes to a line of research studying matroids with no $M(K_4)$-minor, where $M(K_4)$ is the graphic matroid of the complete graph $K_4$.
For example, Sims \cite{Sims-1977} proved that the class of $M(K_4)$-minor-free matroids is complete, Brylawski \cite{Brylawski-1971} characterized the binary matroids with no $M(K_4)$-minor, and Oxley \cite{Oxley-1987} characterized the $3$-connected ternary matroids with no $M(K_4)$-minor.
More recently, work of Pendavingh and van der Pol \cite{Pendavingh-vanderPol-2015} and van der Pol \cite{vanderPol-2023} investigated the importance of $M(K_4)$ in matroid enumeration.
Most notably for this paper, Kung studied Problem \ref{prob: Kung problem} for the class of matroids with no $M(K_4)$-minor and no $U_{2, \ell+2}$-minor, proving that every such simple rank-$r$ matroid has at most $(6\ell^{\ell - 1} + 8\ell)\cdot r$ elements \cite[Theorem 1.1]{Kung-1988}. The graphic matroid $M(K_4)$ is not a positroid, so this also bounds from above the maximum number of elements of a simple rank-$r$ positroid with no $U_{2, \ell+2}$-minor.
To the best of our knowledge, this was in fact the previous best upper bound for this quantity.
There are several other notable $M(K_4)$-minor-free classes of matroids that contain the class of positroids, such as gammoids, strongly base-orderable matroids, and $k$-base-orderable matroids.
We will discuss these further in Section \ref{sec: future work}.

The rest of this paper is structured as follows.
In Section \ref{sec: preliminaries}, we will give more background on matroids and positroids.
In Section \ref{sec: excluded minors}, we will use a list of $9$ known excluded minors for the class of positroids to prove two structural properties for low-rank positroids.
In Section \ref{sec: the extremal examples}, we will describe the matroids for which equality holds in Theorem \ref{thm: main} and prove that they are in fact positroids and are $U_{2, \ell+2}$-minor-free.
We will prove Theorem \ref{thm: main} in Section \ref{sec: the main proof}, and conclude in Section \ref{sec: future work} by discussing several directions for future work.

Our proof of Theorem \ref{thm: main} relies on two new structural properties (Propositions \ref{prop: rank 3 excluded minor} and \ref{prop: rank 4 excluded minor}) for low-rank positroids. The paper then (i) uses a list of $9$ known excluded minors for the class of positroids to prove these two structural properties for low-rank positroids (Section \ref{sec: excluded minors}); (ii) describes the matroids for which equality holds in Theorem \ref{thm: main} and proves that they are positroids and are $U_{2, \ell+2}$-minor-free (Section \ref{sec: the extremal examples}); and (iii) proves Theorem \ref{thm: main} in Section \ref{sec: the main proof}.

Inequality or Bound
Biconditional or Equivalence

Question: Let $r,2 ell\ge 1$ be integers. Recall that a positroid is a matroid representable over $\mathbb{R}$ by a matrix whose maximal minors are all nonnegative, and that $U_{2,m}$ denotes the rank-$2$ uniform matroid on $m$ elements. If $M$ is a simple rank-$r$ positroid with no $U_{2,\ell+2}$-minor, which quantitative estimate for the number of elements $|M|$ holds, and for $r\ge 2$ when does equality occur?

The bound $|M|\le \ell(r-1)+1$ always holds. Moreover, if $r\ge 2$, then equality holds if and only if $M$ can be obtained by taking parallel connections of copies of $U_{2,\ell+1}$.

Choices:
B. The bound $|M|\le \ell r+1$ always holds. Moreover, if $r\ge 2$, then equality holds if and only if $M\cong U_{2,\ell+2}$ or, more generally, $M$ is obtained by taking parallel connections of copies of $U_{2,\ell+2}$.
C. The bound $|M|\le \ell(r-1)+1$ always holds.
D. For each fixed $\ell\ge 1$, there exists an integer $r_0(\ell)$ such that whenever $r\ge r_0(\ell)$ and $M$ is a simple rank-$r$ positroid with no $U_{2,\ell+2}$-minor, one has $|M|\le \ell(r-1)+1$; moreover, for such sufficiently large $r$, equality holds if and only if $M$ can be obtained by taking parallel connections of copies of $U_{2,\ell+1}$.
E. The bound $|M|\le \ell(r-1)+1$ always holds. Moreover, if $r\ge 2$, then equality holds if and only if every long line of $M$ has exactly $\ell+1$ points.

all-ranks statement replaced by sufficiently-large-r

global parallel-connection structure replaced by local long-line condition

The stem does not explicitly reveal the equality characterization, and it does not mention parallel connections or the specific building block $U_{2,\ell+1}$. It states the bound and asks for the equivalent equality case without giving away the correct option.

This is close to theorem recall: the stem presents the extremal bound and asks for the equality characterization. However, it is not a verbatim restatement because the student must still distinguish among several structurally similar conclusions.

Some reasoning is required to reject plausible alternatives such as a direct sum, a weaker minor statement, or the wrong line size. Still, the item mainly tests recognition of the exact characterization rather than substantial derivation.

The distractors are plausible and well-targeted: B confuses parallel connection with direct sum, C is a weaker true-type statement, D overgeneralizes the line sizes, and E incorrectly uses the excluded threshold. These reflect realistic mathematical failure modes.

A solid MCQ with no real answer leakage and strong distractors, but it leans heavily on theorem recall and is only moderately generative rather than deeply reasoning-based.


Question: Let $r,2 ell\ge 1$ be integers. Recall that a positroid is a matroid representable over $\mathbb{R}$ by a matrix whose maximal minors are all nonnegative, and that $U_{2,m}$ denotes the rank-$2$ uniform matroid on $m$ elements. If $M$ is a simple rank-$r$ positroid with no $U_{2,\ell+2}$-minor, which quantitative estimate for the number of elements $|M|$ holds, and for $r\ge 2$ when does equality occur?

Choices:
B. The bound $|M|\le \ell r+1$ always holds. Moreover, if $r\ge 2$, then equality holds if and only if $M\cong U_{2,\ell+2}$ or, more generally, $M$ is obtained by taking parallel connections of copies of $U_{2,\ell+2}$.
C. The bound $|M|\le \ell(r-1)+1$ always holds.
D. For each fixed $\ell\ge 1$, there exists an integer $r_0(\ell)$ such that whenever $r\ge r_0(\ell)$ and $M$ is a simple rank-$r$ positroid with no $U_{2,\ell+2}$-minor, one has $|M|\le \ell(r-1)+1$; moreover, for such sufficiently large $r$, equality holds if and only if $M$ can be obtained by taking parallel connections of copies of $U_{2,\ell+1}$.
E. The bound $|M|\le \ell(r-1)+1$ always holds. Moreover, if $r\ge 2$, then equality holds if and only if every long line of $M$ has exactly $\ell+1$ points.

The stem does not reveal the bound or the equality characterization; it only states the hypotheses and asks for the correct conclusion. There is no explicit answer leakage.

This is essentially a direct theorem-recall item: the stem asks exactly for the quantitative bound and equality case under the stated hypotheses. The correct choice is basically the theorem statement itself.

Some discrimination is required because several choices are close variants (weaker true statement, asymptotic weakening, local-vs-global condition, wrong excluded-minor size). However, solving it mainly depends on remembering the exact theorem rather than generating a conclusion from mathematical reasoning.

The distractors are mathematically plausible and distinct: B uses a natural off-by-one/extremal-family error, C is a weaker true statement, D weakens the quantifier to sufficiently large rank, and E replaces the structural characterization by an insufficient local condition. These align well with realistic failure modes.

A solid theorem-recognition MCQ with strong distractors and no answer leakage, but it scores lower on tautology avoidance and generative reasoning because it mostly asks for the exact recalled statement of a result.


\label{prop:Buo=order} For a sequence $(x_n)$ in a vector lattice $E$ and $x\in E$ the following are equivalent: \begin{enumerate}[label=(\roman*)] \item $x_n\xrightarrow{\Buo} x$; \item $x_n\xrightarrow{\mathrm{o}} x$ (order convergence). \end{enumerate}

In his 1981 paper \cite{Bourgain81}, J.~Bourgain investigated a strengthened
mode of convergence on $\ell_1$: coordinatewise convergence combined with
domination by a single element of $\ell_1$. Bourgain showed, via an elegant
finite-block construction, that this convergence is not sequentially complete,
answering a question posed by Jan Mikusiński.

A natural abstract framework for Bourgain’s phenomenon is provided by vector
lattices. In this setting, coordinatewise convergence is replaced by
\emph{unbounded order} ($\uo$) convergence, while domination is interpreted as
order boundedness by a single positive element. Recall that a sequence
$(x_n)$ in a vector lattice $E$ is said to \emph{\uo-converge} to $x\in E$,
written $x_n\xrightarrow{\uo}x$, if
\[
|x_n-x|\wedge u \xrightarrow{\mathrm{o}} 0
\qquad\text{for every }u\in E_+,
\]
that is, all truncations of $|x_n-x|$ converge to zero in order.

In many familiar settings, $\uo$-convergence coincides with classical pointwise
or almost-everywhere convergence. For example, in $c_0$ and $\ell_p$ it is
exactly coordinatewise convergence, while in standard function spaces it agrees
with almost-everywhere convergence. This leads naturally to the following
definition.

\begin{definition}[Bourgain--uo convergence]
A sequence $(x_n)\subset E$ \emph{\Buo-converges} to $x\in E$ if
$x_n\xrightarrow{\uo}x$ and there exists $y\in E_+$ such that
$|x_n|\leqslant y$ for all $n$.
\end{definition}

Although this definition may appear stronger than order convergence, we show
that for sequences it introduces no genuinely new convergence notion: Bourgain--uo
convergence coincides with ordinary order convergence; only the Cauchy concept we use later is non-standard. We are indebted to Vladimir Troitsky and Mitchell Taylor for pointing this out to us.

\begin{definition}[Bourgain--uo convergence]
A sequence $(x_n)\subset E$ \emph{\Buo-converges} to $x\in E$ if
$x_n\xrightarrow{\uo}x$ and there exists $y\in E_+$ such that
$|x_n|\le y$ for all $n$.
\end{definition}

The novelty lies instead
in the associated Cauchy condition.

\begin{proposition}\label{prop:CX-metric}
Let $(X,d)$ be a metric space with a non-isolated point $x_0\in X$. Then:
\begin{enumerate}[\upshape(i)]
\item if $X$ is compact, the Banach lattice $C(X)$ is not sequentially \Buo-complete;
\item the Banach lattice $C_b(X)$ of bounded continuous functions on $X$ is not sequentially \Buo-complete;
\item if $X$ is locally compact and there exists a decreasing sequence of relatively compact open neighbourhoods $(U_n)$ of $x_0$ with $\bigcap_{n}U_n=\{x_0\}$, then $C_0(X)$ is not sequentially \Buo-complete.
\end{enumerate}
\end{proposition}

We verify directly that $(f_n)$ is \Buo-Cauchy in $C(X)$ (respectively $C_b(X)$).
Let $(n_k)$ be strictly increasing and set $d_k:=f_{n_{k+1}}-f_{n_k}$. For each
$x\in X$ the scalar sequence $(f_n(x))_n$ is decreasing and convergent, hence
$d_k(x)\to0$. Thus $d_k\xrightarrow{\uo}0$ (uo-convergence is pointwise in vector
lattices of real-valued functions). Moreover, $|d_k|\le 1$ for all $k$, and the
constant function $1$ belongs to $C(X)$ and to $C_b(X)$, so the differences are
order bounded by a single element. Hence $d_k\xrightarrow{\Buo}0$, and by
Proposition~\ref{prop:Buo=order} we have $d_k\xrightarrow{\mathrm{o}}0$.
Since $(n_k)$ was arbitrary, $(f_n)$ is \Buo-Cauchy.

\begin{proposition}\label{prop:Lipb-uniformly-discrete}
Let $(X,d)$ be a metric space and let $\mathrm{Lip}_b(X)$ be the vector lattice of bounded
Lipschitz functions $X\to\R$, ordered pointwise. Then $\mathrm{Lip}_b(X)$ is sequentially
\Buo-complete if and only if $\delta(X)>0$.
\end{proposition}

\smallskip\noindent
\emph{Step 3: order convergence.}
For $m\in\N$ define
\[
z_m(x):=\sup_{n\ge m}|f_n(x)-f(x)|,\qquad x\in X.
\]
Then $0\le z_{m+1}\le z_m$ pointwise, and $z_m(x)\downarrow0$ for each $x$ since $f_n(x)\to f(x)$.
Also $z_m(x)\le 2M$ for all $x$, hence $z_m\in\ell_\infty(X)$.
Finally, for every $n\ge m$ we have $|f_n-f|\le z_m$, so $f_n\xrightarrow{\mathrm{o}}f$ in $\ell_\infty(X)$.
By Proposition~\ref{prop:Buo=order}, this is equivalent to $f_n\xrightarrow{\Buo}f$.
Therefore $\ell_\infty(X)$ (and hence $\mathrm{Lip}_b(X)$) is sequentially \Buo-complete.
\end{proof}

\begin{remark}\label{rem:Lipb-characterisation}
Proposition~\ref{prop:Lipb-uniformly-discrete} shows that sequential \Buo-completeness of $\mathrm{Lip}_b(X)$
depends only on the metric geometry of $X$: it holds exactly when $X$ is uniformly discrete.
In particular:
\begin{enumerate}[\upshape(i)]
\item if $X$ has a non-isolated point, then $\delta(X)=0$ and $\mathrm{Lip}_b(X)$ is not sequentially \Buo-complete;
\item if $X$ is discrete but not uniformly discrete (so $d(x)>0$ for all $x$ but $\delta(X)=0$), then $\mathrm{Lip}_b(X)$ is not sequentially \Buo-complete;
\item if $\delta(X)>0$, then $\mathrm{Lip}_b(X)=\ell_\infty(X)$ and $\mathrm{Lip}_b(X)$ is sequentially \Buo-complete.
\end{enumerate}
\end{remark}

\begin{question}\label{q:FBL-lp-sequential}
Let $1<p<\infty$ and $\Gamma$ infinite.
For every \Buo-Cauchy sequence $(f_n)$ in $\mathrm{FBL}[\ell_p(\Gamma)]$,
does the function
\[
z_m(x^\ast)=\sup_{n\ge m}|f_n(x^\ast)-f(x^\ast)|
\]
lie in $\mathrm{FBL}[\ell_p(\Gamma)]$ for each $m$?
Equivalently, must $(f_n)$ converge in order (and hence in the \Buo\ sense)?
\end{question}

\begin{proof} (i)$\Rightarrow$(ii): Suppose $x_n\xrightarrow{\uo}x$ and $|x_n|\leqslant y$ for some $y\in E_+$. Set $w:=(|x|-y)^+\in E_+$. For every $n$ we have $|x_n|\leqslant y$, hence on $\{w>0\}$ we have $|x_n|\leqslant y<|x|$, so \[ |x_n-x|\wedge w \ \geqslant\ (\,|x|-|x_n|\,)\wedge w \ \geqslant\ (|x|-y)\wedge w \ =\ w. \] If $w>0$, this contradicts $|x_n-x|\wedge w\xrightarrow{\mathrm{o}}0$ (take $u=w$ in the definition of $\uo$). Hence $w=0$ and so $|x|\leqslant y$, i.e.\ the sequence $(x_n)$ and its limit $x$ all lie in the order interval $[-y,y]$. On $[-y,y]$ we have \[ |x_n-x| \ =\ |x_n-x|\wedge (2y) \ \leqslant\ 2\bigl(|x_n-x|\wedge y\bigr), \] and since $\uo$-convergence gives $|x_n-x|\wedge y\xrightarrow{\mathrm{o}}0$, we obtain $|x_n-x|\xrightarrow{\mathrm{o}}0$, i.e.\ $x_n\xrightarrow{\mathrm{o}}x$. (ii)$\Rightarrow$(i): If $x_n\xrightarrow{\mathrm{o}}x$, there exists a decreasing sequence $(z_k)\downarrow 0$ with $|x_n-x|\leqslant z_k$ for all $n\geqslant N_k$. In particular the set $\{x_n:n\in\mathbb N\}\cup\{x\}$ is order bounded, so there is $y\in E_+$ with $|x_n|\leqslant y$ for all $n$ (for instance $y:=|x|+\sup_k z_k$). For every $u\in E_+$ and $n\geqslant N_k$, \[ |x_n-x|\wedge u \ \leqslant\ z_k\wedge u\ \downarrow 0, \] so $x_n\xrightarrow{\uo}x$, and thus $x_n\xrightarrow{\Buo}x$ by definition. \end{proof}

The novelty lies instead
in the associated Cauchy condition.

\begin{definition}
A sequence $(x_n)$ in a vector lattice $E$ is called \Buo-\emph{Cauchy} if for
every strictly increasing sequence $(n_k)$ the differences
$x_{n_{k+1}}-x_{n_k}$ converge to $0$ in order in $E$.
\end{definition}

This Cauchy condition is strictly stronger than the usual one induced by order
convergence and captures the oscillatory behaviour underlying Bourgain’s
construction.  The central theme of this paper is to determine which classical
Banach lattices are sequentially complete with respect to this notion.

Our first main result concerns sequence spaces.  We show that both $c_0$ and
$\ell_\infty$ are sequentially \Buo-complete: every \Buo-Cauchy sequence admits
a pointwise limit in the same space and converges to it in order.

We then turn to function lattices.  For bounded Lipschitz functions, the answer
is governed entirely by the metric geometry of the underlying space.  Writing
$\mathrm{Lip}_b(X)$ for the vector lattice of bounded Lipschitz functions on a
metric space $(X,d)$, we prove that $\mathrm{Lip}_b(X)$ is sequentially
\Buo-complete if and only if $X$ is uniformly discrete.  In particular,
$\mathrm{Lip}_b(X)$ fails \Buo-completeness whenever $X$ has a non-isolated point
or, more generally, whenever points of $X$ can accumulate at arbitrarily small
scales.

As further applications, we show that the Banach lattices $C(X)$, $C_b(X)$ and
$C_0(X)$ are not sequentially \Buo-complete under mild topological assumptions
on $X$.  We conclude with several open problems, including the
\Buo-sequential completeness of $\ell_p$ spaces in the reflexive range
$1<p<\infty$ and of free Banach lattices $\mathrm{FBL}[\ell_p(\Gamma)]$.

To prove Proposition~\ref{prop:Buo=order}:

(i)$\Rightarrow$(ii): Assume $x_n\xrightarrow{\uo}x$ and $(x_n)$ is order bounded, i.e. $|x_n|\le y$ for some $y\in E_+$. Set $w:=(|x|-y)^+$. Since $|x_n|\le y$, on $\{w>0\}$ one has $|x_n|\le y<|x|$, hence
\[
|x_n-x|\wedge w\ \ge\ (|x|-|x_n|)\wedge w\ \ge\ (|x|-y)\wedge w\ =\ w.
\]
If $w>0$ this contradicts $|x_n-x|\wedge w\xrightarrow{\mathrm{o}}0$ (take $u=w$ in the definition of $\uo$). Thus $w=0$ and $|x|\le y$, so $x_n,x\in[-y,y]$. On $[-y,y]$,
\[
|x_n-x|=|x_n-x|\wedge (2y)\le 2\bigl(|x_n-x|\wedge y\bigr),
\]
and since $\uo$-convergence gives $|x_n-x|\wedge y\xrightarrow{\mathrm{o}}0$, it follows that $|x_n-x|\xrightarrow{\mathrm{o}}0$, i.e. $x_n\xrightarrow{\mathrm{o}}x$.

(ii)$\Rightarrow$(i): If $x_n\xrightarrow{\mathrm{o}}x$, there exists $(z_k)\downarrow 0$ with $|x_n-x|\le z_k$ for all $n\ge N_k$. Then $\{x_n\}\cup\{x\}$ is order bounded, so choose $y\in E_+$ with $|x_n|\le y$ for all $n$ (e.g. $y:=|x|+\sup_k z_k$). For every $u\in E_+$ and $n\ge N_k$,
\[
|x_n-x|\wedge u\le z_k\wedge u\downarrow 0,
\]
so $x_n\xrightarrow{\uo}x$, and hence $x_n\xrightarrow{\Buo}x$ “by definition.”

To prove the main theorem:

(i)$\Rightarrow$(ii): Assume $x_n\xrightarrow{\uo}x$ and $(x_n)$ is order bounded, i.e. $|x_n|\le y$ for some $y\in E_+$. Set $w:=(|x|-y)^+$. Since $|x_n|\le y$, on $\{w>0\}$ one has $|x_n|\le y<|x|$, hence
\[
|x_n-x|\wedge w\ \ge\ (|x|-|x_n|)\wedge w\ \ge\ (|x|-y)\wedge w\ =\ w.
\]
If $w>0$ this contradicts $|x_n-x|\wedge w\xrightarrow{\mathrm{o}}0$ (take $u=w$ in the definition of $\uo$). Thus $w=0$ and $|x|\le y$, so $x_n,x\in[-y,y]$. On $[-y,y]$,
\[
|x_n-x|=|x_n-x|\wedge (2y)\le 2\bigl(|x_n-x|\wedge y\bigr),
\]
and since $\uo$-convergence gives $|x_n-x|\wedge y\xrightarrow{\mathrm{o}}0$, it follows that $|x_n-x|\xrightarrow{\mathrm{o}}0$, i.e. $x_n\xrightarrow{\mathrm{o}}x$.

(ii)$\Rightarrow$(i): If $x_n\xrightarrow{\mathrm{o}}x$, there exists $(z_k)\downarrow 0$ with $|x_n-x|\le z_k$ for all $n\ge N_k$. Then $\{x_n\}\cup\{x\}$ is order bounded, so choose $y\in E_+$ with $|x_n|\le y$ for all $n$ (e.g. $y:=|x|+\sup_k z_k$). For every $u\in E_+$ and $n\ge N_k$,
\[
|x_n-x|\wedge u\le z_k\wedge u\downarrow 0,
\]
so $x_n\xrightarrow{\uo}x$, and hence $x_n\xrightarrow{\Buo}x$ “by definition.”,

Question: Let $E$ be a vector lattice, let $(x_n)$ be a sequence in $E$, and let $x\in E$. Recall that $(x_n)$ is said to \emph{Bourgain--uo converge} to $x$, written $x_n\xrightarrow{\Buo}x$, if $x_n\xrightarrow{\uo}x$ and there exists $y\in E_+$ such that $|x_n|\le y$ for all $n$, where $x_n\xrightarrow{\uo}x$ means $|x_n-x|\wedge u\xrightarrow{\mathrm{o}}0$ for every $u\in E_+$. Which of the following statements is equivalent to $x_n\xrightarrow{\Buo}x$?

$x_n\xrightarrow{\mathrm{o}}x$; that is, $(x_n)$ converges to $x$ in order in $E$.

Choices:
B. $x_n\xrightarrow{\uo}x$ and there exists, for every $u\in E_+$, an element $y_u\in E_+$ such that $|x_n|\wedge u\le y_u$ for all $n$; equivalently, each truncation $(|x_n|\wedge u)$ is order bounded.
C. $x_n\xrightarrow{\uo}x$ and the set $\{x_n:n\in\mathbb N\}\cup\{x\}$ is order bounded in $E$.
D. $x_n\xrightarrow{\uo}x$ and there exists $y\in E_+$ such that $|x_n-x|\le y$ for all $n$.
E. $x_n\xrightarrow{\uo}x$ and there exists a sequence $(y_m)\subset E_+$ with $y_m\downarrow 0$ such that for every $m$ one has $|x_n|\le y_m$ for all sufficiently large $n$.

replace domination of the sequence alone by order boundedness of the sequence together with the limit

bound on terms instead of bound used to force the limit into the same order interval

eventual vanishing domination replacing fixed order bound

The stem defines Bourgain--uo convergence but does not directly state that it is equivalent to order convergence. The correct option is not leaked explicitly; one must know or derive the relevant theorem.

This is not a verbatim restatement of the definition, since the intended answer invokes a different convergence notion. However, it is still essentially an equivalence-checking theorem question rather than a substantially new inference task.

Some reasoning is required: the solver must connect 'uo + order bounded' with order convergence and distinguish this from nearby variants. But the pressure is weakened because the question is mainly recall of a standard equivalence, and choice C appears so close to the intended theorem that uniqueness is doubtful.

B, D, and E are plausible distractors built from common quantifier and domination confusions. However, C is not a clean distractor: it looks mathematically equivalent or at least extremely close to the intended correct statement, making the item potentially ambiguous.

Moderately good concept-check question with no direct answer leakage, but it is mostly an equivalence-recall item and is weakened by a distractor that appears genuinely true or too close to the correct answer.


\begin{lem}$\mathrm{(Lemma \textbf{ }2.6 \text{ of }\cite{Zhu})}$\label{separation}
		If $ f:M\to M$ is a non-degenerate endomorphism, or, more generally, if $ f:M\to M $ is a local homeomorphism, then $ f $ has uniform separation of preimages.
	\end{lem}

In modern dynamical systems and ergodic theory, both entropy and pressure are the fundamental quantities that measure the complexity of a system. These concepts are closely related to physics and thermodynamic mechanisms; see the monographs by Ruelle \cite{Ruelle} and Bowen \cite{Bowen book}. Roughly speaking, the topological pressure can be interpreted as the maximal free energy that a system can attain. Then, the Gibbs distribution, which can maximize free energy, corresponds to the equilibrium state in ergodic theory. If one neglects the potential energy, topological pressure reduces to topological entropy.

	Invariant measures play central roles in ergodic theory. Let $f$ be a continuous map on a compact metric space $X$. Given an $f$-invariant measure $\mu$, one can define the measure-theoretic entropy $h_{\mu}(f)$. The variational principle establishes a relationship between the measure-theoretic entropy and the topological entropy  $h(f)$ as
	$$
	h(f)=\sup\{h_{\mu}(f) \;|\; \mu\in \mathcal{M}_f\},
	$$
	where $ \mathcal{M}_f $ is the set of $ f $-invariant Borel probability measures.	The element in $ \mathcal{M}_f $ that makes the equality hold is called the \emph{measure of maximal entropy}.
	Given a continuous function $\phi$ on $X$, referred to as a potential, the variational principle for pressure then yields
	$$
	P(f,\phi)=\sup\{h_{\mu}(f)+\int \phi d\mu \;|\; \mu\in \mathcal{M}_f\}.
	$$
	The element in $ \mathcal{M}_f $ that reaches the supremum is called the \emph{equilibrium state} for $\phi$. It follows that when the potential function vanishes, the topological pressure reduces to the topological entropy, and the corresponding equilibrium state coincides with a measure of maximal entropy.

	The investigations into the existence and uniqueness of the equilibrium state and the measure of maximal entropy are of interest to the researchers. Among the techniques in this topic,  there are two classical ones established in the  1970s. The first one is using the transfer operator theory, see in \cite{Bowen book} and \cite{Ruelle}. For a topologically transitive locally maximal hyperbolic set $\Lambda$ of a diffeomorphism $f$, or especially, a basic set of an Axiom A diffeomorphism, one can get a Markov partition of $\Lambda$ and then obtain a symbolic extension of $f_\Lambda$. Hence, applying the transfer operator theory for a finite-type symbolic system, one obtains the existence and uniqueness of the equilibrium state for certain potentials. The second one is using the expansivity and the specification property, see Bowen \cite{Bowen74}. Precisely, for a homeomorphism $f$ on a compact metric space which is expansive and has the specification property, Bowen showed that for a potential function satisfying certain conditions (called the Bowen property), there exists a unique equilibrium state. In fact, if $\Lambda$ is a topologically mixing compact locally maximal hyperbolic set
	for a diffeomorphism $f$, then $f_\Lambda$ is expansive and has the specification property. Therefore, for such type of uniformly hyperbolic systems, there are two ways to investigate the existence and uniqueness of the equilibrium state and the measure of maximal entropy.

	In recent years, the study of dynamics of the differentiable systems beyond uniform hyperbolicity, such as non-uniformly hyperbolic systems and partially hyperbolic systems,  has attracted much attention and made a series of important advances. For partially hyperbolic diffeomorphisms, a center direction is allowed in addition to the hyperbolic directions. The presence of the center direction permits a very rich type of structure. Investigating the thermodynamic mechanisms, especially the existence and uniqueness of the equilibrium state, for partially hyperbolic systems require more exquisite techniques.

	Recently, Climenhaga and Thompson \cite{CT21} ingeniously developed the criterion of Bowen \cite{Bowen74} in settings beyond uniform hyperbolicity
	using weakened versions of specification and expansivity. They first considered the case of symbolic systems.
	Rather than assuming specification on the entire symbolic system, they identified a collection of ``good" words on which the specification property holds, while showing that the set of ``bad" words has topological entropy strictly smaller than that of the full system. Using this decomposition, they established the uniqueness of the measure of maximal entropy and the equilibrium state in \cite{CT12} and \cite{CT13}, respectively. Further, Climenhaga and Thompson weakened expansivity by introducing a weaker, non-uniform notion applicable to general compact metric spaces; they also formulate correspondingly weakened, non-uniform versions of specification, expansivity, and the Bowen property in \cite{CT14} and \cite{CT16}. For applications of the method mentioned above, one can see \cite{CFT18} and \cite{CFT19}. The general strategy is to first identify obstructions to expansivity and specification, and then to control their entropy or pressure. Pacifico, F. Yang, and J. Yang \cite{Yang} further refined the conditions in the Climenhaga-Thompson approach, allowing the framework to be applied in settings with fixed parameters.

	From \cite{CT21}, one can observe that for a partially hyperbolic diffeomorphism with one-dimensional center, if every leaf of the stable and unstable foliations is dense, and the supremum of the measure-theoretic entropy over ergodic measures with non-positive or non-negative central Lyapunov exponent is different, then there exists a unique measure of maximal entropy. A similar argument applies to the case of pressure. Mongez and Pacifico \cite{Mongez} substituted the condition on the central Lyapunov exponent used in \cite{CT21} with the requirement that there exists a strict gap between the unstable entropy and the stable entropy of $ f $. The definitions of unstable entropy and stable entropy referenced here are drawn from \cite{Yan21, HHW}.  Liu and Liao  \cite{LL} showed the robustness and uniqueness of equilibrium states for a class of partially hyperbolic with a central direction that admits a dominated splitting into one-dimensional subbundles and H\"{o}lder continuous potentials with not very large oscillation.

	Partially hyperbolic endomorphisms can be viewed as a generalization of partially hyperbolic diffeomorphisms. However, several distinctions arise when attempting to establish analogous results in this broader setting. Due to the non-invertibility of the map, each point may possess multiple preimages, which prevents the direct definition of unstable manifolds in the classical sense. To address this issue, we adopt the framework of inverse limit spaces for partially hyperbolic endomorphisms, building on the foundational work presented in \cite{Qian}. Through the inverse limit space, the unstable manifolds of partially hyperbolic endomorphisms can be rigorously defined, and corresponding notions of unstable entropy and stable entropy have also been introduced in \cite{Wang, Zhu}.   Álvarez and Cantarino \cite{Cantarino} established the existence of equilibrium states for partially hyperbolic endomorphisms with one-dimensional center bundles by showing that the shift map on the inverse limit space is $h$-expansive, as well as the uniqueness of such states for certain endomorphisms defined on the torus. Recently, Arbieto and  Cabezas  \cite{AC} established the existence of equilibrium states for a class of partially hyperbolic endomorphisms when the central direction is simple or which is decomposed into one-dimensional subbundles with dominated splitting, and gave the finiteness of measures of maximal entropy for a certain partially hyperbolic endomorphism with one-dimensional central direction.

	Our main purpose in this paper is to generalize the results in \cite{Mongez} to the case of partially hyperbolic endomorphisms. Here we assume that the map $ f $ is non-degenerate and thus $ f $ has uniform separation of preimages (see Lemma \ref{separation}), then we can deal with the preimages on a small domain. Denote  $U_a(\phi):=\{\psi\in C(X)\;|\; \lVert \phi-\psi \rVert <a\}$, where $a>0 $ and $C(X)$ represents all continuous functions on $X$ and $ \lVert \phi \rVert=\sup_{x\in X}\lvert \phi(x)\rvert $.

	\begin{mainthm}\label{Main}
		Let $ f:M\to M $ be a non-degenerate $ C^2 $  partially hyperbolic endomorphism on a compact Riemannian manifold $ M $ with one-dimensional center bundle and $ \phi:M\to \mathbb{R} $ be a H\"{o}lder continuous potential. Assume that  $ P^u(f,\phi)>P^s(f,\phi)$ and  the unstable foliation $\tilde{ \mathcal{W}}^u $ of $f$ is minimal, then for any $ 0<a<\frac{P^u(f,\phi)-P^s(f,\phi)}{2} $
		there exists a $ C^1 $ neighborhood $ \mathcal{U}_f $ of $ f $ such that for any H\"{o}lder continuous potential $\psi\in U_{a}(\phi)$ and each non-degenerate $ C^2 $ partially hyperbolic endomorphism $ g\in \mathcal{U}_f $ there   has a unique equilibrium state for $(g,\psi)$.
	\end{mainthm}

The investigations into the existence and uniqueness of the equilibrium state and the measure of maximal entropy are of interest to the researchers. Among the techniques in this topic, there are two classical ones established in the 1970s. The first one is using the transfer operator theory, see in \cite{Bowen book} and \cite{Ruelle}. For a topologically transitive locally maximal hyperbolic set $\Lambda$ of a diffeomorphism $f$, or especially, a basic set of an Axiom A diffeomorphism, one can get a Markov partition of $\Lambda$ and then obtain a symbolic extension of $f_\Lambda$. Hence, applying the transfer operator theory for a finite-type symbolic system, one obtains the existence and uniqueness of the equilibrium state for certain potentials. The second one is using the expansivity and the specification property, see Bowen \cite{Bowen74}. Precisely, for a homeomorphism $f$ on a compact metric space which is expansive and has the specification property, Bowen showed that for a potential function satisfying certain conditions (called the Bowen property), there exists a unique equilibrium state. In fact, if $\Lambda$ is a topologically mixing compact locally maximal hyperbolic set
 for a diffeomorphism $f$, then $f_\Lambda$ is expansive and has the specification property. Therefore, for such type of uniformly hyperbolic systems, there are two ways to investigate the existence and uniqueness of the equilibrium state and the measure of maximal entropy.

From \cite{CT21}, one can observe that for a partially hyperbolic diffeomorphism with one-dimensional center, if every leaf of the stable and unstable foliations is dense, and the supremum of the measure-theoretic entropy over ergodic measures with non-positive or non-negative central Lyapunov exponent is different, then there exists a unique measure of maximal entropy. A similar argument applies to the case of pressure. Mongez and Pacifico \cite{Mongez} substituted the condition on the central Lyapunov exponent used in \cite{CT21} with the requirement that there exists a strict gap between the unstable entropy and the stable entropy of $ f $. The definitions of unstable entropy and stable entropy referenced here are drawn from \cite{Yan21, HHW}. Liu and Liao \cite{LL} showed the robustness and uniqueness of equilibrium states for a class of partially hyperbolic with a central direction that admits a dominated splitting into one-dimensional subbundles and H\"{o}lder continuous potentials with not very large oscillation.

Partially hyperbolic endomorphisms can be viewed as a generalization of partially hyperbolic diffeomorphisms. However, several distinctions arise when attempting to establish analogous results in this broader setting. Due to the non-invertibility of the map, each point may possess multiple preimages, which prevents the direct definition of unstable manifolds in the classical sense. To address this issue, we adopt the framework of inverse limit spaces for partially hyperbolic endomorphisms, building on the foundational work presented in \cite{Qian}. Through the inverse limit space, the unstable manifolds of partially hyperbolic endomorphisms can be rigorously defined, and corresponding notions of unstable entropy and stable entropy have also been introduced in \cite{Wang, Zhu}. Álvarez and Cantarino \cite{Cantarino} established the existence of equilibrium states for partially hyperbolic endomorphisms with one-dimensional center bundles by showing that the shift map on the inverse limit space is $h$-expansive, as well as the uniqueness of such states for certain endomorphisms defined on the torus. Recently, Arbieto and Cabezas \cite{AC} established the existence of equilibrium states for a class of partially hyperbolic endomorphisms when the central direction is simple or which is decomposed into one-dimensional subbundles with dominated splitting, and gave the finiteness of measures of maximal entropy for a certain partially hyperbolic endomorphism with one-dimensional central direction.

Our main purpose in this paper is to generalize the results in \cite{Mongez} to the case of partially hyperbolic endomorphisms. Here we assume that the map $ f $ is non-degenerate and thus $ f $ has uniform separation of preimages (see Lemma \ref{separation}), then we can deal with the preimages on a small domain. Denote $U_a(\phi):=\{\psi\in C(X)\;|\; \lVert \phi-\psi \rVert <a\}$, where $a>0 $ and $C(X)$ represents all continuous functions on $X$ and $ \lVert \phi \rVert=\sup_{x\in X}\lvert \phi(x)\rvert $.

\begin{mainthm}\label{Main}
 Let $ f:M\to M $ be a non-degenerate $ C^2 $ partially hyperbolic endomorphism on a compact Riemannian manifold $ M $ with one-dimensional center bundle and $ \phi:M\to \mathbb{R} $ be a H\"{o}lder continuous potential. Assume that $ P^u(f,\phi)>P^s(f,\phi)$ and the unstable foliation $\tilde{ \mathcal{W}}^u $ of $f$ is minimal, then for any $ 0<a<\frac{P^u(f,\phi)-P^s(f,\phi)}{2} $
 there exists a $ C^1 $ neighborhood $ \mathcal{U}_f $ of $ f $ such that for any H\"{o}lder continuous potential $\psi\in U_{a}(\phi)$ and each non-degenerate $ C^2 $ partially hyperbolic endomorphism $ g\in \mathcal{U}_f $ there has a unique equilibrium state for $(g,\psi)$.
 \end{mainthm}

The following results in \cite{CT16} and \cite{Yang} respectively proved the uniqueness of equilibrium states for certain systems.
 \begin{thm} $(\mathrm{Theorem \text{ }5.6 \text{ of } \cite{CT16}})$\label{CT}
 Let $ X $ be a compact metric space, $ f:X\to X $ a homeomorphism and $ \phi:X\to \mathbb{R} $ a continuous potential function. Suppose that there are $ \varepsilon>0$, $\delta>0 $ with $ \varepsilon>40\delta $ such that $P^\perp_{\mathrm{exp}}(f,\phi,\varepsilon)<P(f,\phi) $ and there exists $ \mathcal{D}\subset X\times \mathbb{N} $ admits a decomposition $ (\mathcal{P},\mathcal{G},\mathcal{S}) $ with the following properties:
 \begin{flushleft}
 (1) For every $ N\in\mathbb{N} $, $ \mathcal{G}^N $ has tail $ (W) $-specification at scale $ \delta $;\\
 (2) $ \phi $ has the Bowen property at scale $ \varepsilon $ on $ \mathcal{G} $;\\
 (3) $P(\mathcal{P}\cup\mathcal{S}\cup\mathcal{D}^c,\phi,\delta,\varepsilon)<P(f,\phi) $.\\
 Then there exists a unique equilibrium state for $ (f,\phi) $.
 \end{flushleft}
 \end{thm}
 \begin{thm} $(\mathrm{Theorem \text{ }A} \text{ of }\cite{Yang})$\label{2.2}
 Let $ X $ be a compact metric space, $ f:X\to X $ a homeomorphism and $ \phi:X\to \mathbb{R} $ a continuous potential function. Suppose that there exist $ \varepsilon>0$, $\delta>0 $ with $ \varepsilon\geq 2000\delta $ such that $ P^\perp_{\mathrm{exp}}(f,\phi,\varepsilon)<P(f,\phi) $ and $ X\times \mathbb{N} $ admits a decomposition $ (\mathcal{P},\mathcal{G},\mathcal{S})$ with the following properties:
 \begin{flushleft}
 (1) $ \mathcal{G} $ has $ (W) $-specification at scale $ \delta $;\\
 (2) $ \phi $ has the Bowen property at scale $ \varepsilon $ on $ \mathcal{G} $;\\
 (3) $ P(\mathcal{P}\cup\mathcal{S},\phi,\delta,\varepsilon)<P(f,\phi) $.\\
 Then there exists a unique equilibrium state for $ (f,\phi) $.
 \end{flushleft}
 \end{thm}
 Theorem \ref{CT} is sometimes referred to as the Climenhaga-Thompson Criterion (abbrev. CT Criterion). As an application of the above criteria, Mongez and Pacifico \cite{Mongez} gave the following result for certain partially hyperbolic diffeomorphisms.
 \begin{thm}$(\mathrm{Theorem \text{ }A} \text{ of }\cite{Mongez})$ \label{Mon}
 Let $f: M\to M$ be a $C^{1+}$ partially hyperbolic diffeomorphism of a compact manifold $M$ with $TM=E^u\oplus E^c\oplus E^s$ and $\phi:M\to \mathbb{R}$ a H\"{o}lder continuous potential. Assume that dim($E^c$)=1 and the unstable foliation $\mathcal{F}^u(f)$ is minimal. If $h^u(f)-h^s(f)>\sup\phi-\inf\phi\geq 0$, then there exists a $C^1$ neighborhood $\mathcal{U}$ of $f$ so that $(g,\phi)$ has a unique equilibrium state for every $C^{1+}$ diffeomorphism $g\in \mathcal{U}$.
 \end{thm}
 The main task of this paper is to obtain the counterpart of Theorem \ref{Mon} in the setting of partially hyperbolic endomorphisms under weaker conditions.

Next, we will use the results in \cite{Yun, Chung, Gelfert} that for any $ C^{1+\alpha} $ mapping, there exists a horseshoe whose topological pressure can approximate the sum of the metric entropy and the integral of the potential whenever the metric entropy is positive for an ergodic hyperbolic measure.

As we know that the horseshoe for $ C^{1+\alpha} $ mapping is structurally stable under small $ C^1 $ perturbation in \cite{Chung}, i.e. there is a $ C^1 $ neighborhood $ \mathcal{U}_1 $ of $ f $ such that there exists a hyperbolic horseshoe $ H_g $ and $
 \tau|_{(H_g)^g}: (H_g)^g \to (H_g)^g $ is topologically conjugate to $\tau|_{(H_\varepsilon)^f}: (H_\varepsilon)^f \to (H_\varepsilon)^f $ by $ h_g:(H_g)^g \to (H_\varepsilon)^f $ with $ \lVert h_g-id \rVert<\delta $
 for any $ g\in \mathcal{U}_1 $ where $(H_g)^g$, $(H_\varepsilon)^f$ are the inverse limit spaces corresponding to the horseshoes respectively and $ \lVert \phi^*\circ h_g-\phi^*\rVert<\varepsilon $ if $ \lVert h_g-id \rVert<\delta $. Then we have
 \begin{align*}
 P(\tau|_{(H_{\varepsilon})^f},\phi^*|_{(H_{\varepsilon})^f})=P(\tau|_{(H_g)^g}, \phi^*\circ h_g).
 \end{align*}
 and
 \begin{align*}
 P^u(f,\phi)-2\varepsilon<P(g|_{H_g}, \phi|_{H_g}).
 \end{align*}
 In order to relate the above inequality to the unstable pressure of $g$, we need to establish the existence of the equilibrium state for $g|_{H_g}$ as follows.
 \begin{claim}
 If $ H_g $ is a hyperbolic set for the endomorphism $ g $, then $ \tau|_{(H_g)^g} $ is expansive.
 \end{claim}
 In other words, we claim that there is a constant $ \eta>0 $ small enough such that for any $ \tilde{x},\tilde{y} \in (H_g)^g$ with $ \tilde{x}\neq \tilde{y}$ then $ \tilde{d}(\tau^k\tilde{x}, \tau^k\tilde{y})>\eta $ for some $ k\in \mathbb{Z} $. Otherwise $ \tilde{y}\in \tilde{W}^u_{\eta}(\tilde{x})\cap \tilde{W}^s_{\eta}(\tilde{x})$.
 Since $ \tilde{x}\in \tilde{W}^u_{\eta}(\tilde{x})\cap \tilde{W}^s_{\eta}(\tilde{x})$ and $ \Pi_{\tilde{W}^u_{loc}(\tilde{x})}: \tilde{W}^u_{loc}(\tilde{x}) \to W^u_{loc}(\tilde{x}) $ is bijective, then by the local product structure in Theorem IV.2.3 of \cite{Qian}, we have $\tilde{x}=\tilde{y} $. This completes the proof of this claim.

Now we assume that $f $ is a $ C^2 $ non-degenerate partially hyperbolic endomorphism on the compact Riemannian manifold with dim$(E^c)=1 $ and $ \phi $ is a H\"{o}lder continuous potential function on $ M $. Recall that let $\phi^*=\phi\circ \Pi$. Given an ergodic measure $ \mu\in \mathcal{M}^e_f $, we can consider the central Lyapunov exponents by Theorem I.2.1 and Proposition I.3.5 of \cite{Qian}. We use $\tilde{\mu} $ to denote the ergodic $ \tau $-invariant Borel probability measure on $ M^f $ such that $ \Pi \tilde{\mu}=\mu $. We denote $ \lambda^c(\tilde{\mu},\tau) $ (resp. $\lambda^c(\mu,f)$) be the central Lyapunov exponent of $ \tilde{\mu} $ (resp. $ \mu$) whenever $ \tilde{\mu}$ (resp. $\mu $) is ergodic, i.e. for $\tilde{\mu}$-$a.e.$ $ \tilde{x}=(x_n)_{n\in\mathbb{Z}}\in M^f$ (resp. $ \mu$-$a.e.$ $x_0\in M $), we have
 \begin{align*}
 \lambda^c(\mu,f)=\lambda^c(\tilde{\mu},\tau)
 &=\lim\limits_{m\to\infty}\frac{1}{m}\log\lvert D_{x_0}f^m \xi\rvert\\
 &=\lim\limits_{m\to\infty}\frac{1}{m}\log\lVert D_{x_0}f^m |_{E^c(\tilde{x})}\rVert\\
 &=\lim\limits_{m\to\infty}\frac{1}{m}\log\lVert D_{x_{m-1}}f\circ \cdots \circ D_{x_1}f \circ D_{x_0}f |_{E^c(\tilde{x})}\rVert\\
 &=\lim\limits_{m\to\infty}\frac{1}{m}\sum_{k=0}^{m-1}\log\lVert D_{x_k}f |_{E^c(\tau^k\tilde{x})}\rVert\\
 &=\int_{M^f} \log\lVert Df |_{E^c(\tilde{x})}\rVert d\tilde{\mu},
 \end{align*}
 where $ \tilde{x}\in M^f $ with $ \Pi(\tilde{x})=x_0 $, $ 0\neq\xi\in E^c(\tilde{x}) $, and define $\tilde{\varphi}^c(\tilde{x}):=\log\lVert Df |_{E^c(\tilde{x})}\rVert$.

\begin{lem}
 There exists $ r>0 $ such that
 $ \sup\{h_{\tilde{\mu}}(\tau)+\int\phi^* d\tilde{\mu}: \tilde{\mu}\in \mathcal{M}_\tau, \lambda^c(\tilde{\mu},\tau)\geq -r\}<P(\tau,\phi^*) $.
 \end{lem}
 \begin{pf}
 First, we claim that
 $ \sup\{h_{\tilde{\mu}}(\tau)+\int\phi^* d\tilde{\mu}: \tilde{\mu}\in \mathcal{M}_\tau, \lambda^c(\tilde{\mu},\tau)\geq 0\}<P(\tau,\phi^*) $. If not, we can assume there exists an invariant probability measure $ \tilde{\mu} $ with $ \lambda^c(\tilde{\mu},\tau)\geq 0 $, but $ \tilde{\mu} $ is an equilibrium state for $ (\tau,\phi^*) $. By the ergodic decomposition theorem, we have
 $ \lambda^c(\tilde{\mu},\tau)=\int \tilde{\varphi}^c d\tilde{\mu}=\int_{\mathcal{M}_\tau^e} (\int \tilde{\varphi}^c dm) d\tau(m) \geq 0$,
 and by Theorem 8.4 in \cite{Walters} we get
 $ h_{\tilde{\mu}}(\tau)=\int_{\mathcal{M}_\tau^e} h_m(\tau)d\tau(m) $.
 Then
 $$
 P(\tau,\phi^*)=h_{\tilde{\mu}}(\tau)+\int \phi^* d\tilde{\mu}
 =\int_{\mathcal{M}_\tau^e} (h_m(\tau)+\int \phi^* dm)d\tau(m),
 $$
 so each $ m\in \mathcal{M}_\tau^e $ must be the equilibrium state for $ (\tau,\phi^*) $. Therefore, there exists a measure $ m_0\in\mathcal{M}_\tau^e $ such that
 $ \lambda^c(m_0,\tau)=\int \tilde{\varphi}^c dm_0\geq 0 $
 and
 $ P(\tau,\phi^*)=h_{m_0}(\tau)+\int \phi^* dm_0 $,
 which contradicts with $ P^+(\tau,\phi^*)<P^-(\tau,\phi^*) $.

The proof of the main theorem in this paper is inspired by arguments established in \cite{CT21} and \cite{Mongez}. Compared with the case of diffeomorphisms, the main difficulty here comes from non-invertibility. In order to overcome this deficiency, we use the technique of inverse limit. For certain non-degenerate partially hyperbolic endomorphisms with one-dimensional center and any  H\"{o}lder continuous potential function satisfy specific conditions, we aim to construct a decomposition of orbit segments in the inverse limit spaces. An important condition in the main theorem is the gap between the unstable pressure and the stable pressure. For partially hyperbolic diffeomorphisms, the unstable entropy and unstable pressure for a continuous function were introduced in \cite{HHW, HWZ}, respectively. As a generalization, the concepts of unstable entropy and unstable pressure were given by Wang, Wu and Zhu in \cite{Wang} for partially hyperbolic endomorphisms. In \cite{Zhu}, Wu and Zhu introduced the stable entropy for non-degenerate $C^1$  partially hyperbolic endomorphisms and their stable pressure is given in \cite{Li}. Different from the condition given by Mongez and Pacifico in \cite{Mongez}, we change the condition ``$h^u(f)-h^s(f)>\sup\phi-\inf\phi\geq 0$'' to ``$P^u(f,\phi)>P^s(f,\phi)$'', where
	$P^u(f,\phi)=\sup\{h^u_{\mu}(f)+\int \phi d\mu\;|\; \mu\in \mathcal{M}_f^e\}$ and $P^s(f,\phi)=\sup\{h^s_{\mu}(f)+\int \phi d\mu\;|\; \mu\in \mathcal{M}_f^e\}$.
	Since for each ergodic $f$-invariant measure $\mu$, there is
	\begin{align*}
		h^u_\mu(f)+\min\phi \leq h^u_\mu(f)+\int \phi d\mu,
	\end{align*}
	we have $P^u(f,\phi)\geq h^u(f)+\min\phi$.
	Meanwhile, $h^s(f)+\max\phi \geq P^s(f,\phi)$. Therefore, the condition ``$P^u(f,\phi)>P^s(f,\phi)$'' is weaker than ``$h^u(f)-h^s(f)>\sup\phi-\inf\phi\geq 0$'' since the space we are considering is compact so $\sup\phi = \max\phi$ and $\inf\phi=\min\phi$.
	Under the condition $ P^u(f,\phi)>P^s(f,\phi) $ and assuming the existence of unstable foliation with some sense of minimality, we show that the specification property holds for good orbit segments that exhibit a contracting property in the center-stable direction. Moreover, the potential $ \phi $ satisfies the Bowen property on these good orbit segments. Additionally, the topological pressure on the collection of bad orbit segments and the pressure associated with the obstruction to expansivity are both strictly less than the topological pressure of the entire system. As a consequence, there exists a unique equilibrium state on the inverse limit space, and its projection yields the unique equilibrium state for the original partially hyperbolic endomorphism. Furthermore, we also demonstrate that the conditions of the theorem are inherently robust, which further implies the robustness of the equilibrium state.
	At the same time, H\"{o}lder continuous potential functions can also vary within a certain range,  thus allowing the uniqueness of the equilibrium state of the system under small perturbation can be obtained.

	This paper is organized as follows. In Section 2, we give some preliminaries, including the concepts of the decomposition of the orbit segments, the pressure of the obstructions to expansivity, partially hyperbolic endomorphisms, inverse limit space, (un)stable en        tropy and (un)stable pressure, and other necessary definitions. In Section 3, we demonstrate the robustness of the conditions in Main Theorem. In Section 4, we prove  Main Theorem using the improved Climenhaga-Thompson criterion. In Section 5, we give an example satisfying the conditions in our main theorem.

The proof is said to be “inspired by arguments established in \cite{CT21} and \cite{Mongez},” with the “main difficulty” (vs. diffeomorphisms) coming from “non-invertibility,” which is handled by “the technique of inverse limit.” Under the condition $P^u(f,\phi)>P^s(f,\phi)$ (introduced as a weakening of Mongez–Pacifico’s condition), the strategy is to “construct a decomposition of orbit segments in the inverse limit spaces.” Assuming “the existence of unstable foliation with some sense of minimality,” they “show that the specification property holds for good orbit segments that exhibit a contracting property in the center-stable direction,” and that the potential $\phi$ “satisfies the Bowen property on these good orbit segments.” They also show that “the topological pressure on the collection of bad orbit segments and the pressure associated with the obstruction to expansivity are both strictly less than the topological pressure of the entire system.” Then, “as a consequence, there exists a unique equilibrium state on the inverse limit space,” and “its projection yields the unique equilibrium state for the original partially hyperbolic endomorphism.” Finally, they “demonstrate that the conditions of the theorem are inherently robust,” implying “robustness of the equilibrium state,” and note that H\"older potentials “can also vary within a certain range,” giving uniqueness “under small perturbation.”

The results of the Main Theorem still hold for $ C^{1+\alpha} $ mappings. To cite some conclusions in \cite{Qian} conveniently, we therefore assume $ C^2 $ regularity here.,

Question:
Let $M$ be a compact Riemannian manifold, let $f:M\to M$ be a non-degenerate $C^{1+\alpha}$ partially hyperbolic endomorphism for some $\alpha>0$ with one-dimensional center bundle, and let $\phi:M\to\mathbb{R}$ be a H\"older continuous potential. Assume that the unstable pressure and stable pressure satisfy $P^u(f,\phi)>P^s(f,\phi)$, and that the unstable foliation $\tilde{\mathcal W}^u$ (defined on the inverse-limit space associated to $f$) is minimal, meaning every unstable leaf is dense. For $a>0$, define
$$U_a(\phi):=\{\psi\in C(M):\|\phi-\psi\|_\infty<a\}.$$
For any
$$0<a<\frac{P^u(f,\phi)-P^s(f,\phi)}{2},$$
which statement holds for every such $f$, $\phi$, and $a$?

There exists a $C^1$ neighborhood $\mathcal U$ of $f$ such that for every $g\in\mathcal U$ and every $\psi\in U_a(\phi)$, the system $g$ admits a unique equilibrium state for the potential $\psi$.

Choices:
B. There exists a $C^1$ neighborhood $\mathcal U$ of $f$ such that for every $g\in\mathcal U$ and every $\psi\in U_a(\phi)$, the system $g$ admits at least one equilibrium state for the potential $\psi$, and this equilibrium state is unique whenever $P^u(g,\psi)>P^s(g,\psi)$.
C. There exists a $C^1$ neighborhood $\mathcal U$ of $f$ such that the system $g$ admits a unique equilibrium state for the potential $\phi$ for every $g\in\mathcal U$.
D. There exists a $C^1$ neighborhood $\mathcal U$ of $f$ such that for every $g\in\mathcal U$ and every $\psi\in U_a(\phi)$, the system $g$ admits a unique equilibrium state for the potential $\psi$ whenever $0<a<P^u(f,\phi)-P^s(f,\phi)$.
E. For every $\psi\in U_a(\phi)$, there exists a $C^1$ neighborhood $\mathcal U_\psi$ of $f$ such that for every $g\in\mathcal U_\psi$, the system $g$ admits a unique equilibrium state for the potential $\psi$.

robust uniqueness tied to fixed pressure-gap/minimality hypotheses rather than rechecking perturbed pressures

dropped the uniformity over all nearby potentials $\psi\in U_a(\phi)$

half-gap bound on allowable oscillation radius

uniform neighborhood independent of the perturbed potential

The stem does not explicitly state the conclusion or identify choice A. Although the hypotheses are tailored to the target theorem, the correct answer is not directly leaked from the wording alone.

This is essentially a theorem-recall item: the stem gives the exact hypotheses and asks for the conclusion that holds. Choice A is basically the theorem statement, with distractors formed by small perturbations of it.

There is some reasoning required to distinguish the exact quantifiers and the sharp half-gap condition from nearby false variants, but the main task is recognizing the precise theorem conclusion rather than generating a new argument.

The distractors are plausible and mathematically meaningful: they vary uniformity in the neighborhood, dependence on the perturbed potential, and the sharpness of the pressure-gap bound. These align with common failure modes in reading technical statements.

A solid theorem-discrimination MCQ with strong distractors, but it is close to a direct restatement of the source theorem, so it tests recall/statement parsing more than genuinely generative reasoning.