\begin{theorem}\label{thm1}
There exists a constant $C>1$ such that
\begin{equation}\label{equnthm1}
C^{-1}\lambda^{d_S/2}\leq{N}(u)\leq C\lambda^{d_S/2}
\end{equation}
holds for any global Dirichlet or Neumann eigenfunction $u$  on $\mathcal {SG}$ with  eigenvalue $\lambda$, except for the first non-constant Neumann eigenfunction where $N(u)=0$.
\end{theorem}

\label{sec1}
The study of Laplacians on fractals and their spectral properties constitutes a cornerstone of analysis on fractals, a field largely pioneered by Kigami \cite{K2,K4}. Unlike their Euclidean counterparts, Laplacians on fractals exhibit a wealth of novel and unexpected phenomena, leading to a rich and distinct spectral theory \cite{FS, Ka1,Ka2,K3,KL, Sh, Sh2, T}. Among these phenomena, the intricate oscillatory behavior of eigenfunctions — such as the distribution and growth of their local extrema — remains a topic of profound interest.

In the classical setting of a smooth, compact Riemannian manifold (or a bounded Euclidean domain with suitable boundary conditions), the celebrated Courant nodal domain theorem \cite{CH} provides a fundamental upper bound on the number of nodal domains of an eigenfunction, which in turn implies an upper bound for its local extrema. Moreover, the Hörmander-type estimates \cite{Ho} show that the $L^\infty$-norm of an eigenfunction grows at most polynomially with the eigenvalue. These results paint a picture of eigenfunctions whose complexity increases in a controlled and predictable manner as the energy (eigenvalue) increases.

The landscape on fractals is quite different. The existence of pre-localized eigenfunctions — eigenfunctions that vanish identically on the boundary along with their normal derivatives — is a hallmark of fractals \cite{K3,K4}. Such eigenfunctions can be highly localized and give rise to  a cascade of new eigenfunctions through a localization process. This structure fundamentally alters the asymptotic distribution of eigenvalues and the qualitative behavior of eigenfunctions. Consequently, classical tools and intuition from elliptic PDEs often fail, necessitating new frameworks for understanding the fine properties of eigenfunctions on fractals.

\begin{figure}[h]
\begin{center}
\includegraphics[width=5.5cm]{SG2.pdf}
\caption{The Sierpinski gasket $\mathcal{SG}$.}
\label{figure1}
\end{center}
\end{figure}

This paper is devoted to a study of the oscillatory behavior of eigenfunctions on post-critically finite (p.c.f.) self-similar sets, among which the Sierpinski gasket ($\mathcal{SG}$, see Figure \ref{figure1}) serves as a typical  example. We introduce and investigate the extremum counting function
$N(x)$ for the Laplacian $-\Delta_\mu$ where $\mu$ is the reference measure. For an eigenfunction $u$ with eigenvalue $\lambda$, the count
$N(u)$ enumerates its distinct local maximum and minimum sets (formally defined in Definition \ref{def1}). The function $N(x)$ then captures the maximum of this number over all eigenfunctions with eigenvalues up to $x$:

$$N(x)=\sup\{N(u): u \text{ is a }\lambda \text{-eigenfunction with } 0\leq \lambda\leq x\}.$$

Our primary goal is to establish the asymptotic growth rate of $N(x)$ as $x\to \infty$, linking it directly to the spectral exponent $d_S$ of the fractal, which characterizes a sharp phase transition in the behavior of $N(x)$.

For the upper bound, we prove in Proposition \ref{prothm1} that under a natural condition (A) — namely, that eigenfunctions with sufficiently small eigenvalues possess at most one extreme set — the growth of $N(x)$ is at most of the order $x^{d_S/2}$:
$$\limsup_{x\to\infty} \frac{N(x)}{x^{d_S/2}}<\infty.$$
Condition (A) can be interpreted as a form of ``low-energy simplicity''. It is not universal for all fractals, as evidenced by the modified Koch curve \cite{M,Sh2}, where even low-energy eigenfunctions can have infinitely many extrema. Verifying its validity is therefore a key step in the analysis for a given fractal.

Conversely, for the lower bound, Proposition \ref{prothm2} shows that the existence of  a  pre-localized eigenfunction forces \( N(x) \) to grow at least polynomially. In the lattice case, this lower bound is sharp, matching the upper bound:
\[
\liminf_{x\to\infty}\frac{N(x)}{x^{d_S/2}}>0.
\]
The proof constructs sums of copies of a pre-localized eigenfunction over appropriately chosen cells — an inherently fractal method with no direct analogue in the smooth setting.

Our main result  is the following theorem, which gives a precise, uniform, two-sided estimate of $N(u)$ for eigenfunctions on the Sierpinski gasket $\mathcal{SG}$.

In the classical setting of a smooth, compact Riemannian manifold (or a bounded Euclidean domain with suitable boundary conditions), the celebrated Courant nodal domain theorem \cite{CH} provides a fundamental upper bound on the number of nodal domains of an eigenfunction, which in turn implies an upper bound for its local extrema. Moreover, the Hörmander-type estimates \cite{Ho} show that the $L^\infty$-norm of an eigenfunction grows at most polynomially with the eigenvalue. These results paint a picture of eigenfunctions whose complexity increases in a controlled and predictable manner as the energy (eigenvalue) increases.

\begin{figure}[h]
\begin{center}
\includegraphics[width=5.5cm]{SG2.pdf}
\caption{The Sierpinski gasket $\mathcal{SG}$.}
\label{figure1}
\end{center}
\end{figure}

This paper is devoted to a study of the oscillatory behavior of eigenfunctions on post-critically finite (p.c.f.) self-similar sets, among which the Sierpinski gasket ($\mathcal{SG}$, see Figure \ref{figure1}) serves as a typical example. We introduce and investigate the extremum counting function
$N(x)$ for the Laplacian $-\Delta_\mu$ where $\mu$ is the reference measure. For an eigenfunction $u$ with eigenvalue $\lambda$, the count
$N(u)$ enumerates its distinct local maximum and minimum sets (formally defined in Definition \ref{def1}). The function $N(x)$ then captures the maximum of this number over all eigenfunctions with eigenvalues up to $x$:

$$N(x)=\sup\{N(u): u \text{ is a }\lambda \text{-eigenfunction with } 0\leq \lambda\leq x\}.$$

Conversely, for the lower bound, Proposition \ref{prothm2} shows that the existence of a pre-localized eigenfunction forces \( N(x) \) to grow at least polynomially. In the lattice case, this lower bound is sharp, matching the upper bound:
\[
\liminf_{x\to\infty}\frac{N(x)}{x^{d_S/2}}>0.
\]
The proof constructs sums of copies of a pre-localized eigenfunction over appropriately chosen cells — an inherently fractal method with no direct analogue in the smooth setting.

Our main result is the following theorem, which gives a precise, uniform, two-sided estimate of $N(u)$ for eigenfunctions on the Sierpinski gasket $\mathcal{SG}$.

\begin{definition}\label{def1}

(a).
Let $u\in\mathscr{D}_\mu$, $\emptyset\neq A\subset K$ and $c\in\mathbb{R}$, we say that $u$ has a \textit{local maximum ({\rm resp.} minimum)} at $A$ with value $c$ if

(a-1). $A$ is a connected component of $u^{-1}(c)$;

(a-2). $A\cap V_0=\emptyset$;

(a-3). there exists $\delta>0$ such that $u(p)\leq c$ (resp. $u(p)\geq c$) for any $ p\in A_\delta$, where
$A_\delta$ is the $\delta$-neighborhood of $A$.

Call such a set $A$ a \textit{(local) extreme set} of $u$. Further, if $A$ is a singleton, call $A$ an {\it extreme point} of $u$.

(b). For $u\in\mathscr{D}_\mu$, denote
\[{N}(u)=\#\{A\subset K: A \text{ is an extreme set of } u\}.\]
For $x>0$, define
\[N(x)=\sup\{{N}(u): u \text { is a $\lambda$-eigenfunction with}\ 0\leq \lambda\leq x\},\]
and call it \textit{the extremum counting function} of $-\Delta_\mu$.
\end{definition}

\begin{proposition}\label{prothm2}
If there exists a pre-localized eigenfunction, then
\begin{equation*}
\liminf_{x\to\infty}\frac{\log N(x)}{\log x}\geq\kappa
\end{equation*}
for some $0<\kappa\leq d_S/2$.
In particular, for the lattice case (see the exact meaning in Section \ref{sec2}),
\begin{equation*}
\liminf_{x\to\infty}\frac{N(x)}{x^{d_S/2}}>0.
\end{equation*}
\end{proposition}

\noindent{\bf Remark.}
For a $\lambda$-eigenfunction with $\lambda>0$, the values of its local maxima (resp. minima) are positive (resp. negative).
Indeed, suppose that $u$ has a local maximum at $A$ with value $c\leq 0$. Since $u$ is a $\lambda$-eigenfunction, there exists $\delta>0$ such that $u\leq c\leq 0$ and $\Delta_\mu u=-\lambda u\geq 0$ on $A_\delta$. Moreover, $u$ is non-constant on $A_\delta$. Let $h$ be the harmonic function satisfying $h|_{\partial A_\delta}=u|_{\partial A_\delta}$, and denote by $g(\cdot,\cdot)$ the Green function for $-\Delta_\mu$ on $A_\delta$. It is known that $g(p,q)>0$ for all $p,q\in A_\delta$. Then, for $p\in A$, we have
\begin{equation*}
u(p)=\int_{A_\delta}g(p,q)\lambda u(q)\mathrm{d}\mu(q)+h(p)<h(p)\leq \max u|_{\partial A_{\delta}},
\end{equation*}
which yields a contradiction.

\noindent{\bf Remark.} By Lemma \ref{pro1}-(a), $\mathbf{A}_\lambda$ and $\mathbf{B}_\lambda^1$ are cones, i.e. if $\mathbf{a}\in \mathbf{A}_\lambda\ (\text{or }\mathbf{B}_\lambda^1)$, then $k\mathbf{a}\in \mathbf{A}_\lambda \ (\text{or }\mathbf{B}_\lambda^1)$ for any $k\neq 0$. Let $\pi:\mathbb{R}^L\setminus\{\mathbf{0}\}\to\mathbb{R}\mathrm{P}^{L-1}$ be the canonical projection that sends each point to the line through the origin it generates.
For $\mathbf{a},\mathbf{b}\in\mathbb{R}^L$, define $\mathbf{a}\sim\mathbf{b}$ if there exists $k\in\mathbb{R}\setminus\{0\}$ such that $\mathbf{a}=k\mathbf{b}$, then $\pi$ can be regarded as a bijection from $(\mathbb{R}^L\setminus\{\mathbf{0}\})/\sim$ to $\mathbb{R}\mathrm{P}^{L-1}$.
Write \[\mathcal{A}_\lambda=\pi(\mathbf{A}_\lambda),\quad \mathcal{B}_\lambda^1=\pi(\mathbf{B}_\lambda^1),\] and define
\begin{equation}\label{defabT}
\mathcal{T}_\lambda^i:\mathbb{R}\mathrm{P}^{L-1}\to\mathbb{R}\mathrm{P}^{L-1} \quad \text{by}\quad \mathcal{T}_\lambda^i=\pi\circ T_\lambda^i\circ\pi^{-1}.
\end{equation}
Then the condition (\ref{eq1}) is equivalent to
\begin{equation}\tag{\ref{eq1}*}\label{eq2}
\begin{aligned}
&\mathcal{A}_\lambda=\big(\bigcup_{i\in S}(\mathcal{T}_\lambda^i)^{-1}(\mathcal{A}_{\gamma^2_i\lambda})\big)\cup\mathcal{B}^1_\lambda,\quad\text{with}\\
(\mathcal{T}_\lambda^i&)^{-1}(\mathcal{A}_{\gamma^2_i\lambda}),\ i\in S\text{ and }\mathcal{B}^1_\lambda\text{ pairwise disjoint.}
\end{aligned}
\end{equation}
Consequently, condition (A) is equivalent to \eqref{eq2}.

First we prove \eqref{exC}, i.e. $\mathcal{C}_{\lambda}=\mathcal{D}_{\lambda_0}$. Noticing that for any $\mathbf{b}\in\mathbb{R}^3\setminus\{{\bf 0}\}$, by \eqref{eq4.4}, we have
\begin{equation}\label{pixxi}
\pi(\{\mathbf{x}\in\mathbb{R}^3\setminus\{{\bf 0}\}:\mathbf{b}^\mathrm{t}\mathbf{x}=0\})=\Big\{\boldsymbol{\xi}\in\mathbb{R}\mathrm{P}^2:\ \mathbf{b}^\mathrm{t}(\mathbf{1},Q)^{-\mathrm{t}}\begin{pmatrix}1\\ \boldsymbol{\xi}\end{pmatrix}=0\Big\},
\end{equation}
and a direct calculation gives
\begin{equation*}
(4-\lambda_0,-2,-2)(\mathbf{1},Q)^{-\mathrm{t}}=\frac{1}{3}\big(-\lambda_0,0,2(6-\lambda_0)\big).
\end{equation*}
So by Lemma \ref{pro43}, we have
\begin{equation*}
\begin{aligned}
&\pi\circ\tau_\lambda^D\circ(\tau_\lambda^N)^{-1}\Big(\{(x^{(1)},x^{(2)},x^{(3)})^\mathrm{t}\in\mathbb{R}^3\setminus\{{\bf 0}\}:x^{(1)}=0\}\Big)\\
=&\Big\{\boldsymbol{\xi}\in\mathbb{R}\mathrm{P}^2:\ \big(-\lambda_0,0,2(6-\lambda_0)\big)\begin{pmatrix}1\\ \boldsymbol{\xi}\end{pmatrix}=0\Big\}\\
=&\Big\{(\xi^{(1)},\xi^{(2)})^\mathrm{t}\in\mathbb{R}^2:\ \xi^{(2)}-\frac{\lambda_0}{2(6-\lambda_0)}=0\Big\}\cup\{[(1,0)^\mathrm{t}]_\infty\}.
\end{aligned}
\end{equation*}
Similarly, $(-2,4-\lambda_0,-2)(\mathbf{1},Q)^{-\mathrm{t}}=\frac{1}{3}\big(-\lambda_0,-\sqrt{3}(6-\lambda_0),-(6-\lambda_0)\big)$ gives
\begin{equation*}
\begin{aligned}
&\pi\circ\tau_\lambda^D\circ(\tau_\lambda^N)^{-1}\Big(\{(x^{(1)},x^{(2)},x^{(3)})^\mathrm{t}\in\mathbb{R}^3\setminus\{{\bf 0}\}:x^{(2)}=0\}\Big)\\
=&\{(\xi^{(1)},\xi^{(2)})^\mathrm{t}\in\mathbb{R}^2:\ \sqrt{3}\xi^{(1)}+\xi^{(2)}+\frac{\lambda_0}{6-\lambda_0}=0\}\cup\{[(1,-\sqrt{3})^\mathrm{t}]_\infty\},
\end{aligned}
\end{equation*}
and $(-2,-2,4-\lambda_0)(\mathbf{1},Q)^{-\mathrm{t}}=\frac{1}{3}\big(-\lambda_0,\sqrt{3}(6-\lambda_0),-(6-\lambda_0)\big)$ gives
\begin{equation*}
\begin{aligned}
&\pi\circ\tau_\lambda^D\circ(\tau_\lambda^N)^{-1}\Big(\{(x^{(1)},x^{(2)},x^{(3)})^\mathrm{t}\in\mathbb{R}^3\setminus\{{\bf 0}\}:x^{(3)}=0\}\Big)\\
=&\{(\xi^{(1)},\xi^{(2)})^\mathrm{t}\in\mathbb{R}^2:\ -\sqrt{3}\xi^{(1)}+\xi^{(2)}+\frac{\lambda_0}{6-\lambda_0}=0\}\cup\{[(1,\sqrt{3})^\mathrm{t}]_\infty\}.
\end{aligned}
\end{equation*}
Recall that $\mathbf{C}=\{(a_1,a_2,a_3)\in\mathbb{R}^3:\text{either }a_i>0\text{ for all }i, \text{ or }a_i<0\text{ for all }i\}$ and $\mathcal{C}_\lambda=\pi\circ\tau_\lambda^D\circ(\tau_\lambda^N)^{-1}(\mathbf{C})$. By the symmetry of $\mathcal{SG}$, we have $\boldsymbol{\theta}\in\mathcal{C}_\lambda$. So the above calculation gives
$\mathcal{C}_{\lambda}=\mathcal{D}_{\lambda_0}$.

(b). For $5^{-m}\lambda<\lambda_1^D$ with $m\geq 1$, it is easy to see that
$\lambda_{m}=\psi^{-1}(5^{-m}\lambda)\in(0,6)$ by Remark \ref{re33}. Considering the function $v=u\circ F_2F_3^{m-1}$, by (\ref{gamma-delta01pre}), (\ref{gamma-delta02pre}) and $p_{23}=F_2F_3^{m-1}p_3\in A$, we see
\begin{equation}\label{gamma-delta0102r}
\begin{aligned}
&2v(p_1)+2v(p_2)-(4-\lambda_m)v(p_3)=0,\\
&\big(v(p_{3})-v(p_2)\big)\big(v(p_{3})-v(p_1)\big)\geq 0.
\end{aligned}
\end{equation}
Using (\ref{pixxi}) we find that
\begin{equation*}
\begin{aligned}
&\tau(v)\in\pi\circ\tau_\lambda^D\Big(\big\{v\in E(5^{-m}\lambda)\setminus\{0\}: v\text{ satisfies (\ref{gamma-delta0102r})}\big\}\Big)\\
=&\ \big\{(\xi^{(1)},\xi^{(2)})\in\mathbb{R}^2:\xi^{(2)}=\sqrt{3}\xi^{(1)}-\frac{\lambda_m}{6-\lambda_m},\ 0\leq\xi^{(1)}\leq\frac{\sqrt{3}\lambda_m}{2(6-\lambda_m)}\big\}\\
=&\ \mathcal{I}_{\lambda_m,3}\cup\{\boldsymbol{\zeta}_{\lambda_m,23}\}\cup\{\boldsymbol{\zeta}_{\lambda_m,31}\},
\end{aligned}
\end{equation*}
since
\begin{equation*}
\begin{aligned}
&(2,2,-(4-\lambda_m))(\mathbf{1},Q)^{-\mathrm{t}}=\frac{1}{3}\big(\lambda_m,-\sqrt{3}(6-\lambda_m),6-\lambda_m\big),\\
&(0,-1,1)(\mathbf{1},Q)^{-\mathrm{t}}=\frac{1}{3}\big(0,2\sqrt{3},0\big),\\
&(-1,0,1)(\mathbf{1},Q)^{-\mathrm{t}}=\frac{1}{3}\big(0,\sqrt{3},-3\big),
\end{aligned}
\end{equation*}
and $\pi\circ \tau_\lambda^D(v')=\frac{\lambda_m}{4(6-\lambda_m)}(\sqrt{3},-1)^\mathrm{t}\in\mathcal{I}_{\lambda_m,3}$ for a function $v'\in E(5^{-m}\lambda)$ satisfying (\ref{gamma-delta0102r}) with $v'(p_1)=v'(p_2)=4-\lambda_m$ and $v'(p_3)=4$.

(b). for $u$ in the (D5) or (N5) case, there exists a constant $c_5>1$ such that
\begin{equation}\label{esti5}
c_5^{-1}\lambda^{d_S/2}\leq N(u)\leq c_5\lambda^{d_S/2};
\end{equation}

\begin{lemma}\label{lm62}
Let $u\in E_D(\lambda)\setminus\{0\}$ or $E_N(\lambda)\setminus\{0\}$ with $\mathrm{supp}\,u=\mathcal{SG}$. For $u$ in the (D6) or (N6) case and $\eps=\varnothing$, we have
\begin{equation}\label{esti6}
c_6^{-1}\lambda^{d_S/2}\leq N(u)\leq c_6\lambda^{d_S/2}
\end{equation}
for some constant $c_6>1$.
\end{lemma}
\begin{proof}
For $u$ in the (D6) or (N6) case, we have $\lambda_m=3$, $\lambda_{m-1}=6$ and $\lambda=5^m\psi(3)$. Now we consider $u\circ F_{w'}$ for each $w'\in W_{m-2}$. Since $-\Delta_{m-1}u|_{V_{m-1}}=\lambda_{m-1}u|_{V_{m-1}}$, for distinct $i,j,k\in S$ we have
\begin{equation*}
\begin{aligned}
-2u(p_{ij}^{w'})=u(p_{i}^{w'})+u(p_{j}^{w'})+u(p_{ki}^{w'})+u(p_{jk}^{w'}),
\end{aligned}
\end{equation*}
and solving the above equations gives $u(p_i^{w'})=-u(p_{ki}^{w'})-u(p_{ij}^{w'})$. Without loss of generality, we assume $u(p_{23}^{w'})\leq u(p_{31}^{w'})\leq u(p_{12}^{w'})$, then

The result demonstrates the highly regular behavior of  $\mathcal{SG}$, where eigenfunction complexity — quantified by the count of local extrema — grows  precisely as the  power law  $d_S/2$. The proof relies on a detail analysis on the eigenfunction decimation, originated developed by Rammal and Toulouse \cite{RT} and later rigorously established by Shima and Fukushima \cite{Sh, FS}.

In the classical setting, the number of critical points of an eigenfunction on an $n$-dimensional manifold is generally expected to grow linearly with $\lambda^{(n-1)/2}$, a rate related to the wave propagation and the Bohr-Sommerfeld quantization. However, in highly  symmetric setting such as $n$-dimensional rectangles or balls — where eigenfunctions can oscillate at high frequencies independently along different directions — this order improves to $\lambda^{n/2}$.

On fractals, as shown  in Theorem \ref{thm1} and Propositions \ref{prothm1} and \ref{prothm2}, the growth is governed by the spectral exponent $d_S$, which takes the role played by the geometric Hausdorff dimension $n$ in the Euclidean setting. The observed order $d_S/2$ for $\mathcal{SG}$ suggests that its high symmetry is a key factor in realizing this maximal growth rate. We therefore conjecture that for a broader class of p.c.f. self-similar sets, the growth order of local extrema is at most $d_S/2$, with this upper bound being attained only in highly symmetric cases  — such as nested fractals, or even beyond the p.c.f. setting, as exemplified by the  Sierpinski carpet — and strictly smaller in the presence of lower symmetry.

\subsection{Notation and Propositions \ref{prothm1} and \ref{prothm2}}Before ending this section, let us introduce the exact definition of   $N(u)$ and $N(x)$  on a p.c.f. self-similar set $K$, which is always assumed to be  connected.

Let $V_0$ denote the boundary of $K$, and $\mathscr{D}_\mu$ denote the domain of $\Delta_\mu$. For a function $u\in \mathscr{D}_\mu$ and $p\in V_0$, denote by $(du)_p$ the normal derivative of $u$ at $p$.
For $\lambda\geq 0$, call a non-trivial  function $u\in\mathscr{D}_\mu$ satisfying $-\Delta_\mu u=\lambda u$ on $K\setminus V_0$ a {\it $\lambda$-eigenfunction} of $-\Delta_\mu$.  Say an eigenfunction $u$ a \textit{Dirichlet (Neumann) eigenfunction} if  $u|_{V_0}=0$ ($du|_{V_0}=0$). In particular, say $u$ a \textit{pre-localized eigenfunction}  if  both $u|_{V_0}=0$ and $du|_{V_0}=0$ hold; and a \textit{global eigenfunction} if $\mathrm{supp}\,u=K$.

\begin{definition}\label{def1}

(a).
Let $u\in\mathscr{D}_\mu$, $\emptyset\neq A\subset K$ and $c\in\mathbb{R}$, we say that $u$ has a \textit{local maximum ({\rm resp.} minimum)} at $A$ with value $c$ if

(a-1). $A$ is a connected component of $u^{-1}(c)$;

(a-2). $A\cap V_0=\emptyset$;

(a-3). there exists $\delta>0$ such that $u(p)\leq c$ (resp. $u(p)\geq c$) for any $ p\in A_\delta$, where
$A_\delta$ is the $\delta$-neighborhood of $A$.

Call such a set $A$ a \textit{(local) extreme set} of $u$. Further, if $A$ is a singleton, call $A$ an {\it extreme point} of $u$.

(b). For $u\in\mathscr{D}_\mu$, denote
\[{N}(u)=\#\{A\subset K: A \text{ is an extreme set of } u\}.\]
For $x>0$, define
\[N(x)=\sup\{{N}(u): u \text { is a $\lambda$-eigenfunction with}\ 0\leq \lambda\leq x\},\]
and call it \textit{the extremum counting function} of $-\Delta_\mu$.
\end{definition}

\noindent{\bf Remark.}
For a $\lambda$-eigenfunction with $\lambda>0$, the values of its local maxima (resp. minima) are positive (resp. negative).
Indeed, suppose that $u$ has a local maximum at $A$ with value $c\leq 0$. Since $u$ is a $\lambda$-eigenfunction, there exists $\delta>0$ such that $u\leq c\leq 0$ and $\Delta_\mu u=-\lambda u\geq 0$ on $A_\delta$. Moreover, $u$ is non-constant on $A_\delta$. Let $h$ be the harmonic function satisfying $h|_{\partial A_\delta}=u|_{\partial A_\delta}$, and denote by $g(\cdot,\cdot)$ the Green function for $-\Delta_\mu$ on $A_\delta$. It is known that $g(p,q)>0$ for all $p,q\in A_\delta$. Then, for $p\in A$, we have
\begin{equation*}
u(p)=\int_{A_\delta}g(p,q)\lambda u(q)\mathrm{d}\mu(q)+h(p)<h(p)\leq \max u|_{\partial A_{\delta}},
\end{equation*}
which yields a contradiction.

We have certain asymptotic estimates of $N(x)$ as $x\to \infty$.

\noindent{\bf(A)}: there exists $\lambda_0>0$ such that ${N}(u)\leq 1$ for any $\lambda$-eigenfunction $u$ with $0\leq\lambda<\lambda_0$.

\begin{proposition}\label{prothm1}
If condition (A) holds, then
\begin{equation*}
\limsup_{x\to\infty}\frac{N(x)}{x^{d_S/2}}< \infty.
\end{equation*}
\end{proposition}

\begin{proposition}\label{prothm2}
If there exists a pre-localized eigenfunction, then
\begin{equation*}
\liminf_{x\to\infty}\frac{\log N(x)}{\log x}\geq\kappa
\end{equation*}
for some $0<\kappa\leq d_S/2$.
In particular, for the lattice case (see the exact meaning in Section \ref{sec2}),
\begin{equation*}
\liminf_{x\to\infty}\frac{N(x)}{x^{d_S/2}}>0.
\end{equation*}
\end{proposition}

We structure the paper as follows.

In Section \ref{sec2}, we present the proofs of  Propositions \ref{prothm1} and \ref{prothm2}, and provide an equivalent characterization of condition (A).

Beginning in Section \ref{sec3}, we focus on the canonic Laplacian on the Sierpinski gasket $\mathcal{SG}$. There, we recall the spectral decimation method and state two key preparatory theorems — Theorems \ref{thm4} and \ref{thm5}.  The former confirms the validity of condition (A), while the later establishes two-sided estimate of $N(u)$ for a special class of eigenfunctions.

The proof of  Theorem \ref{thm4} is given in Section \ref{sec4},  followed by the proof of Theorem \ref{thm5} in Section \ref{sec5}. We conclude in Section \ref{sec6} with the proof of Theorem \ref{equnthm1}.

The post-theorem introduction says that the proof of Theorem~\ref{thm1} “relies on a detail analysis on the eigenfunction decimation,” i.e. the spectral (eigenfunction) decimation method “originated developed by Rammal and Toulouse \cite{RT} and later rigorously established by Shima and Fukushima \cite{Sh, FS}.” It further indicates the overall organization supporting Theorem~\ref{thm1}: in Section~\ref{sec3} the paper “recall[s] the spectral decimation method” and states “two key preparatory theorems — Theorems~\ref{thm4} and \ref{thm5},” where “\ref{thm4} confirms the validity of condition (A), while the later establishes two-sided estimate of $N(u)$ for a special class of eigenfunctions,” and then “conclude[s] in Section~\ref{sec6} with the proof” of the main estimate \eqref{equnthm1} (i.e. Theorem~\ref{thm1}).

The post-theorem introduction says that the proof of the main theorem “relies on a detail analysis on the eigenfunction decimation,” i.e. the spectral (eigenfunction) decimation method “originated developed by Rammal and Toulouse \cite{RT} and later rigorously established by Shima and Fukushima \cite{Sh, FS}.” It further indicates the overall organization supporting the main theorem: next the paper “recall[s] the spectral decimation method” and states “two key preparatory theorems.” We first record the following theorem.

\begin{theorem}\label{thm4}
For $0<\lambda<\lambda_1^D$, we have
\begin{equation}\label{eq3}
\begin{aligned}
\mathcal{A}_\lambda=\ & \mathcal{C}_\lambda=\mathcal{D}_{\psi^{-1}(\lambda)},\quad \mathcal{B}_\lambda^1=\big(\bigcup_{ij\in S_1}\mathcal{L}_{\psi^{-1}(\lambda),ij}\big)\cup\{\boldsymbol{\theta}\},\quad\text{and}\\
&(\mathcal{T}_\lambda^i)^{-1}(\mathcal{A}_{5^{-1}\lambda})=
\mathcal{G}_{\psi^{-1}(\lambda),i}\quad\text{for }i\in S.
\end{aligned}
\end{equation}
In particular, condition (A) holds.
\end{theorem}

The introduction then points to the later preparatory result
\begin{theorem}\label{thm5}
For  $\eps\in\{-1,1\}^n$ of length $n\geq 1$ with $\eps_n=1$, we have
\begin{equation}\label{equn}
c^{-1}(\lambda^{\eps})^{d_S/2}\leq N(u^{\eps})\leq c(\lambda^{\eps})^{d_S/2}
\end{equation}
with some constant $c>1$ independent of $\lambda_0$, $\mathbf a$ and $\eps$.
\end{theorem}

and finally says it “conclude[s] later with the proof” of the main estimate
\begin{equation}\label{equnthm1}
C^{-1}\lambda^{d_S/2}\leq{N}(u)\leq C\lambda^{d_S/2}
\end{equation}
(i.e. the main theorem).

Existential–Universal
Inequality or Bound

Question: Let \(\mathcal{SG}\) be the Sierpinski gasket, and let \(u\) be a global Dirichlet or global Neumann eigenfunction of the Laplacian on \(\mathcal{SG}\) with eigenvalue \(\lambda\). For \(u\), define \(N(u)\) to be the number of its extreme sets, where an extreme set is a connected component \(A\) of a level set \(u^{-1}(c)\), disjoint from the boundary set \(V_0\), such that for some \(\delta>0\), either \(u(p)\le c\) for all \(p\) in the \(\delta\)-neighborhood of \(A\) (local maximum case) or \(u(p)\ge c\) for all such \(p\) (local minimum case). Let \(d_S\) denote the spectral dimension of \(\mathcal{SG}\). Which quantitative estimate holds for \(N(u)\)?

There exists a constant \(C>1\) such that for every global Dirichlet or Neumann eigenfunction \(u\) on \(\mathcal{SG}\) with eigenvalue \(\lambda\), one has
\[
C^{-1}\lambda^{d_S/2}\le N(u)\le C\lambda^{d_S/2},
\]
except for the first non-constant Neumann eigenfunction, for which \(N(u)=0\).

Choices:
B.
There exists a constant \(C>1\) such that for every global Dirichlet or Neumann eigenfunction \(u\) on \(\mathcal{SG}\) with eigenvalue \(\lambda\), one has
\[
C^{-1}\lambda^{d_S/2}\le N(u)\le C\lambda^{d_S/2},
\]
with no exceptional case.
C.
There exists a constant \(C>1\) such that for every global Dirichlet or Neumann eigenfunction \(u\) on \(\mathcal{SG}\) with eigenvalue \(\lambda\), one has
\[
N(u)\le C\lambda^{d_S/2},
\]
except for the first non-constant Neumann eigenfunction, for which \(N(u)=0\).
D.
For every global Dirichlet or Neumann eigenfunction \(u\) on \(\mathcal{SG}\) with eigenvalue \(\lambda\), there exists a constant \(C=C(u)>1\) such that
\[
C^{-1}\lambda^{d_S/2}\le N(u)\le C\lambda^{d_S/2},
\]
except for the first non-constant Neumann eigenfunction, for which \(N(u)=0\).
E.
There exists a constant \(C>1\) such that for every global Dirichlet or Neumann eigenfunction \(u\) on \(\mathcal{SG}\) with eigenvalue \(\lambda\), one has
\[
C^{-1}\lambda\le N(u)\le C\lambda,
\]
except for the first non-constant Neumann eigenfunction, for which \(N(u)=0\).

The stem does not state or strongly hint at the correct estimate; it only defines the objects and asks which quantitative bound is valid.

This is essentially a direct theorem-recall item: the student is asked to identify the exact quantitative estimate that holds, with the correct option closely matching the theorem statement.

There is some pressure to distinguish the exact strongest valid statement from nearby variants (missing exception, weaker bound, wrong exponent, wrong quantifier), but the task is still mostly recognition/recall rather than substantive derivation.

The distractors are plausible and mathematically targeted: they test common failure modes such as omitting an exceptional case, weakening a two-sided estimate, altering uniformity of constants, or using the wrong growth exponent.

A solid theorem-statement discrimination question with strong distractors, but it is primarily recall-based and only weakly tests generative reasoning.


\begin{definition}[Strong non-integrability condition]    \label{def:strong non-integrability condition}
    Let $f \colon S^2 \mapping S^2$ be an expanding Thurston map and $d$ be a visual metric on $S^2$ for $f$.
    Fix $\holderexp \in (0, 1]$.
    Let $\potential \in \holderspacesphere$ be a real-valued \holder continuous function with an exponent $\holderexp$.
    \begin{enumerate}[label=\rm{(\arabic*)}]
        \smallskip
        \item We say that $\potential$ satisfies the \emph{$(\mathcal{C}, \holderexp)$-strong non-integrability condition} (with respect to $f$ and $d$), for a Jordan curve $\mathcal{C} \subseteq S^2$ with $\post{f} \subseteq \mathcal{C}$, if there exist
        \begin{enumerate}[label=\rm{(\alph*)}]
            \smallskip
            \item numbers $\juxtapose{N}{M} \in \n$, $\varepsilon \in (0, 1)$,
            \smallskip
            \item $M$-tiles $Y^{M}_{\black} \in \cTile{M}{\black}$, $Y^{M}_{\white} \in \cTile{M}{\white}$
        \end{enumerate}
        such that for each $\colour \in \colours$, each integer $m \geqslant M$, and each $m$-tile $X \in \Tile{m}$ with $X \subseteq Y^{M}_{\colour}$, there exist two points $\juxtapose{x_{1}}{x_{2}} \in X$ with the following properties:
        \begin{enumerate}
            \smallskip
            \item $\min \set[\big]{ d \parentheses[\big]{ x_{1}, S^2 \mysetminus X }, d \parentheses[\big]{ x_{2}, S^2 \mysetminus X }, d(x_{1}, x_{2}) } \geqslant \diameter{d}{X}$, and
            \smallskip
            \item for each integer $n \geqslant N$, there exist two $(n + M)$-tiles $\juxtapose{X^{n + M}_{\colour, 1}}{X^{n + M}_{\colour, 2}} \in \Tile{n + M}$ such that $Y^{M}_{\colour} = f^{n} \parentheses[\big]{ X^{n + M}_{\colour, 1} } = f^{n} \parentheses[\big]{ X^{n + M}_{\colour, 2} }$ and
            \[
                \abs{ S_{n}\potential(\varsigma_{1}(x_{1})) - S_{n}\potential(\varsigma_{2}(x_{1})) - S_{n}\potential(\varsigma_{1}(x_{2})) + S_{n}\potential(\varsigma_{2}(x_{2})) } \geqslant \varepsilon d( x_{1}, x_{2})^{\holderexp},
            \]
            where we write $\varsigma_{i} \define \parentheses[\big]{ f^{n}|_{X^{n + M}_{\colour, i}} }^{-1}$ for each $i \in \set{1, 2}$.
        \end{enumerate}
        \smallskip

        \item We say that $\potential$ satisfies the \emph{$\holderexp$-strong non-integrability condition} (with respect to $f$ and $d$) if $\potential$ satisfies the $(\mathcal{C}, \holderexp)$-strong non-integrability condition with respect to $f$ and $d$ for some Jordan curve $\mathcal{C} \subseteq S^2$ with $\post{f} \subseteq \mathcal{C}$.

        \smallskip

        \item We say that $\potential$ satisfies the \emph{strong non-integrability condition} (with respect to $f$ and $d$) if $\potential$ satisfies the $\holderexp'$-strong non-integrability condition with respect to $f$ and $d$ for some $\holderexp' \in (0, \holderexp]$.
    \end{enumerate}
\end{definition}

\begin{definition}[Eventually positive function]    \label{def:eventually positive functions}
    Let $g \colon X \mapping X$ be a map on a set $X$, and $\varphi \colon X \mapping \real$ be a real-valued function on $X$.
    Then $\varphi$ is \emph{eventually positive} if there exists $N \in \n$ such that $S_n \varphi(x) > 0$ for each $x \in X$ and each $n \in \n$ with $n \geqslant N$.
\end{definition}

\begin{equation}  \label{eq:Variational Principle for entropy}
    h_{\operatorname{top}}(g) = \sup\{h_{\mu}(g) \describe\mu \in \mathcal{M}(X, g)\}.
\end{equation}
A measure $\mu$ that attains the supremum in \eqref{eq:Variational Principle for pressure} is called an \emph{equilibrium state} for the map $g$ and the potential $\psi$. A measure $\mu$ that attains the supremum in \eqref{eq:Variational Principle for entropy} is called a \emph{measure of maximal entropy} of $g$.

Let $\widetilde{X}$ be another compact metric space. If $\mu$ is a measure on $X$ and the map $\pi \colon X \mapping \widetilde{X}$ is continuous, then the \emph{push-forward} $\pi_{*} \mu$ of $\mu$ by $\pi$ is the measure given by $\pi_{*}\mu(A) \define \mu \parentheses[\big]{ \pi^{-1}(A) }$ for all Borel sets $A \subseteq \widetilde{X}$.

\subsection{Thurston maps}\label{sub:Thurston_maps}

In this subsection, we go over some key concepts and results on Thurston maps, and expanding Thurston maps in particular.
For a more thorough treatment of the subject, we refer to \cite{bonk2017expanding,li2017ergodic}.

Let $S^2$ denote an oriented topological $2$-sphere and $f \colon S^2 \mapping S^2$ be a branched covering map.
We denote by $\deg_f(x)$ the local degree of $f$ at $x \in S^2$.
The \emph{degree} of $f$ is $\deg{f} = \sum_{x\in f^{-1}(y)} \deg_{f}(x)$ for $y\in S^2$ and is independent of $y$.

A point  $x\in S^2$ is a \emph{critical point} of $f$ if $\deg_f(x) \geqslant 2$.
The set of critical points of $f$ is denoted by $\crit{f}$. A point $y\in S^2$ is a \emph{postcritical point} of $f$ if $y = f^n(x)$ for some $x \in \crit{f}$ and $n\in \n$.
The set of postcritical points of $f$ is denoted by $\post{f}$.
We observe that $\post{f} = \post{f^{n}}$ for all $n \in \n$.

\begin{definition}[Thurston maps]
    A Thurston map is a branched covering map $f \colon S^2 \mapping S^2$ on $S^2$ with $\deg f \geqslant 2$ and $\card{ \post{f} }  < +\infty$.
\end{definition}

We now recall the notation for cell decompositions of $S^2$ as used in \cite{bonk2017expanding} and \cite{li2017ergodic}. A \emph{cell of dimension $n$} in $S^2$, $n \in \{1, \, 2\}$, is a subset $c \subseteq S^2$ that is homeomorphic to the closed unit ball $\overline{\mathbb{B}^n}$ in $\real^n$, where $\mathbb{B}^{n}$ is the open unit ball in $\real^{n}$. We define the \emph{boundary of $c$}, denoted by $\partial c$, to be the set of points corresponding to $\partial \mathbb{B}^n$ under such a homeomorphism between $c$ and $\overline{\mathbb{B}^n}$. The \emph{interior of $c$} is defined to be $\inte{c} = c \mysetminus \partial c$. For each point $x\in S^2$, the set $\{x\}$ is considered as a \emph{cell of dimension $0$} in $S^2$. For a cell $c$ of dimension $0$, we adopt the convention that $\partial c = \emptyset$ and $\inte{c} = c$.

Let $f \colon S^2 \mapping S^2$ be a Thurston map, and $\mathcal{C}\subseteq S^2$ be a Jordan curve containing $\post{f}$.
Then the pair $f$ and $\mathcal{C}$ induces natural cell decompositions $\mathbf{D}^n(f,\mathcal{C})$ of $S^2$, for each $n \in \n_0$, in the following way:

By the Jordan curve theorem, the set $S^2 \mysetminus \mathcal{C}$ has two connected components. We call the closure of one of them the \emph{white $0$-tile} for $(f,\mathcal{C})$, denoted by $X^0_{\white}$, and the closure of the other one the \emph{black $0$-tile} for $(f,\mathcal{C})$, denoted be $X^0_{\black}$.
The set of $0$-\emph{tiles} is $\mathbf{X}^0(f, \mathcal{C}) \define \set[\big]{ X^0_{\black}, \, X^0_{\white} }$.
The set of $0$-\emph{vertices} is $\mathbf{V}^0(f, \mathcal{C}) \define \post{f}$.
We set $\overline{\mathbf{V}}^0(f, \mathcal{C}) \define \set[\big]{ \{x\} \describe x\in \mathbf{V}^0(f,\mathcal{C}) }$.
The set of $0$-\emph{edges} $\mathbf{E}^0(f,\mathcal{C})$ consists of the closures of the connected components of $\mathcal{C} \mysetminus \post{f}$.
Then we get a cell decomposition\[
    \mathbf{D}^0(f,\mathcal{C}) \define \mathbf{X}^0(f, \mathcal{C}) \cup \mathbf{E}^0(f,\mathcal{C}) \cup \overline{\mathbf{V}}^0(f,\mathcal{C})
\]
of $S^2$ consisting of \emph{cells of level }$0$, or $0$-\emph{cells}.

We can recursively define the unique cell decomposition $\mathbf{D}^n(f,\mathcal{C})$ for $n\in \n$, consisting of $n$-\emph{cells}, such that $f$ is cellular for $\parentheses[\big]{ \mathbf{D}^{n + 1}(f,\mathcal{C}), \mathbf{D}^n(f,\mathcal{C}) }$.
See \cite[Lemma~5.12]{bonk2017expanding} for details.
We denote by $\mathbf{X}^n(f,\mathcal{C})$ the set of $n$-cells of dimension 2, called $n$-\emph{tiles}; by $\mathbf{E}^n(f,\mathcal{C})$ the set of $n$-cells of dimension $1$, called $n$-\emph{edges}; by $\overline{\mathbf{V}}^n(f,\mathcal{C})$ the set of $n$-cells of dimension $0$; and by $\mathbf{V}^n(f,\mathcal{C})$ the set $\set[\big]{x \describe \{x\} \in \overline{\mathbf{V}}^n(f,\mathcal{C}) }$, called the set of $n$-\emph{vertices}.

\smallskip

For $n\in \n_0$, we define the \emph{set of black $n$-tiles} as\[
    \mathbf{X}^n_{\black}(f,\mathcal{C}) \define \left\{ X \in \mathbf{X}^n (f,\mathcal{C}) \describe f^n(X) = X^0_{\black} \right\},
\]
and the \emph{set of white $n$-tiles} as\[
    \mathbf{X}^n_{\white}(f,\mathcal{C}) \define \left\{X\in \mathbf{X}^n(f,\mathcal{C}) \describe f^n(X) = X^0_{\white}\right\}.
\]

From now on, if the map $f$ and the Jordan curve $\mathcal{C}$ are clear from the context, we will sometimes omit $(f,\mathcal{C})$ in the notation above.

We can now give a definition of expanding Thurston maps.

\begin{definition}[Expansion]     \label{def:expanding_Thurston_maps}
    A Thurston map $f \colon S^2 \mapping S^2$ is called \emph{expanding} if there exists a metric $d$ on $S^2$ that induces the standard topology on $S^2$ and a Jordan curve $\mathcal{C} \subseteq S^2$ containing $\post{f}$ such that
    \begin{equation}    \label{eq:definition of expansion}
        \lim_{n \to +\infty} \max\{ \diameter{d}{X} \describe X \in \mathbf{X}^n(f,\mathcal{C}) \} = 0.
    \end{equation}

\begin{theorem} \label{thm:main theorem}
    Let $f \colon S^2 \to S^2$ be an expanding Thurston map and $d$ be a visual metric on $S^2$ for $f$.
    Let $\beta \in (0, 1]$ and $\phi \in C^{0, \beta}(S^2, d)$ be an eventually positive real-valued \holder continuous function satisfying the $\beta$-strong non-integrability condition (with respect to $f$ and $d$).
    Then there exists a unique positive number $\rootpressure > 0$ with topological pressure $P(f, -\rootpressure \phi) = 0$ and there exists $N_{f} \in \n$ depending only on $f$ such that for each $N \in \n$ with $N \geqslant N_{f}$, the following statement holds for the iterate $F \coloneqq f^{N}$ and the potential $\Phi \coloneqq \sum_{i = 0}^{N - 1} \phi \circ f^{i}$:

    Denote $\alpha \coloneqq \frac{\mathrm{d}}{\mathrm{d} t} P(F, t \Phi) |_{t = -\rootpressure}$ and $\sigma \coloneqq \sqrt{ \frac{\mathrm{d}^2}{\mathrm{d} t^2} P(F, t \Phi) |_{t = -\rootpressure } }$.
    Let $\sequen{I_{n}}$ be a sequence of intervals contained in a compact set $K \subseteq \real$ with $\sequen[\big]{|I_{n}|^{-1}}$ having sub-exponential growth.
    Then
    \[
        \pi_{F, \Phi}(n; \alpha, I_n) \sim \frac{ \int_{I_{n}} e^{\rootpressure t} \, \mathrm{d}t }{ \sqrt{2\pi} \, \sigma } \, \frac{e^{\rootpressure \alpha n}}{n^{3/2}} \qquad \text{as } n \to +\infty.
    \]
\end{theorem}

\begin{remark}\label{rem:chordal metric visual metric qs equiv}
    If $f \colon \ccx \mapping \ccx$ is a rational expanding Thurston map, then a visual metric is quasisymmetrically equivalent to the chordal metric on the Riemann sphere $\ccx$ (see \cite[Theorem~18.1~(ii)]{bonk2017expanding}).
    Here the chordal metric $\sigma$ on $\ccx$ is given by $\sigma (z, w) \define \frac{2\abs{z - w}}{\sqrt{1 + \abs{z}^2} \sqrt{1 + \abs{w}^2}}$ for all $\juxtapose{z}{w} \in \cx$, and $\sigma(\infty, z) = \sigma(z, \infty) \define \frac{2}{\sqrt{1 + \abs{z}^2}}$ for all $z \in \cx$.
    Quasisymmetric embeddings of bounded connected metric spaces are \holder continuous (see \cite[Section~11.1 and Corollary~11.5]{heinonen2001lectures}).
    Accordingly, the classes of \holder continuous functions on $\ccx$ equipped with the chordal metric and on $S^2 = \ccx$ equipped with any visual metric for $f$ are the same (up to a change of the \holder exponent).
\end{remark}

\label{sec:Introduction}

Periodic orbits serve as the skeleton of chaotic dynamics, encoding essential information about the system's long-term behavior.
A fundamental objective in this field is counting these orbits, a problem analogous to the Prime Number Theorem in number theory.
Recently, the dynamical counterpart---the Prime Orbit Theorem---was successfully established for expanding Thurston maps \cite{li2024prime:dirichlet, li2024prime:split}, which serve as topological models for postcritically-finite rational maps.
While this answers the question of ``how many'' orbits exist, it leaves open the more subtle question of ``how they are distributed'' with respect to statistical observables.
Understanding this fine-scale distribution is crucial for characterizing the fluctuations inherent in the system.

In this paper, we investigate the asymptotic distribution of primitive periodic orbits restricted to shrinking intervals for expanding Thurston maps.
We count orbits whose Birkhoff sums for a given potential lie within a prescribed family of shrinking intervals.
we obtain a precise asymptotic formula for the number of these constrained orbits, which resemble a local central limit theorem.
Unlike the central limit theorem, which describes the distribution of Birkhoff sums on the scale of $\sqrt{n}$, the local central limit theorem can probe the density at a given point.
Our main result refines the coarse counting of the Prime Orbit Theorem \cite{li2024prime:dirichlet} for expanding Thurston maps.

The problem of counting orbits in shrinking intervals has been successfully addressed for uniformly expanding systems.
For instance, Petkov and Stoyanov \cite{petkovDistributionPeriodsClosed2012} investigated the distribution of closed orbits for hyperbolic flows, and Sharp and Stylianou \cite{sharpStatisticsMultipliersHyperbolic2022} studied the multipliers and holonomies for hyperbolic rational maps.
However, these results rely heavily on the hyperbolicity and smoothness of the underlying systems.
The context of non-uniformly expanding dynamics, particularly for branched covering maps, remains largely unexplored.
To the best of our knowledge, our work is the first to address this problem in such a setting.

\subsection{Main results}\label{sub:Main results}

Let $f \colon S^2 \to S^2$ be an expanding Thurston map and $\phi \colon S^2 \to \mathbb{R}$ be a real-valued H\"{o}lder  continuous function.
The topological $2$-sphere $S^2$ is equipped with a \emph{visual metric} $d$ (see Section~\ref{sub:Thurston_maps} for details).
A periodic orbit $\tau = \set[\big]{x, f(x), \dots, f^{n-1}(x)}$ (where $f^n(x) = x$) is called \emph{primitive} if $f^m(x) \ne x$ for each integer $m$ with $1 \leqslant m < n$.
We denote the set of primitive periodic orbits of $f$ by $\mathfrak{P}(f)$.
For each $\tau \in \mathfrak{P}(f)$, we write $\phi(\tau) \coloneqq \sum_{x \in \tau} \phi(x)$.

In this article, we investigate the asymptotic distribution of primitive periodic orbits subject to constraints on their Birkhoff sums.
Specifically, for a given number $\alpha \in \mathbb{R}$ and a sequence $\sequen{I_{n}}$ of intervals contained in a compact set $K \subseteq \mathbb{R}$, we study the asymptotic behavior of
\[
    \pi_{f, \phi}(n; \alpha, I_n) \coloneqq \operatorname{card} \set[\big]{ \tau \in \mathfrak{P}_n(f) : \phi(\tau) - n \alpha \in I_{n}}
\]
as $n \to +\infty$, where $\mathfrak{P}_n(f) \coloneqq \set[\big]{\tau \in \mathfrak{P}(f) : |\tau| = n}$.

To obtain precise estimates, we impose specific conditions on the potential $\phi$ and the sequence $\sequen{I_{n}}$.
We assume that $\phi$ is \emph{eventually positive} and satisfies the \emph{strong non-integrability condition} (see Definitions~\ref{def:eventually positive functions} and~\ref{def:strong non-integrability condition}).
Furthermore, denoting the length of $I_{n}$ by $|I_n|$, we assume that the sequence $\sequen[\big]{|I_{n}|^{-1}}$ exhibits sub-exponential growth, i.e., $\limsup_{n \to +\infty} \frac{1}{n} \log \parentheses[\big]{ |I_{n}|^{-1} } = 0$.

We write $A(n) \sim B(n)$ as $n \to +\infty$ if $\lim_{n \to +\infty} A(n) / B(n) = 1$.

Periodic orbits serve as the skeleton of chaotic dynamics, encoding essential information about the system's long-term behavior.
A fundamental objective in this field is counting these orbits, a problem analogous to the Prime Number Theorem in number theory.
Recently, the dynamical counterpart---the Prime Orbit Theorem---was successfully established for expanding Thurston maps \cite{li2024prime:dirichlet, li2024prime:split}, which serve as topological models for postcritically-finite rational maps.
While this answers the question of ``how many'' orbits exist, it leaves open the more subtle question of ``how they are distributed'' with respect to statistical observables.
Understanding this fine-scale distribution is crucial for characterizing the fluctuations inherent in the system.

In this paper, we investigate the asymptotic distribution of primitive periodic orbits restricted to shrinking intervals for expanding Thurston maps.
We count orbits whose Birkhoff sums for a given potential lie within a prescribed family of shrinking intervals.
we obtain a precise asymptotic formula for the number of these constrained orbits, which resemble a local central limit theorem.
Unlike the central limit theorem, which describes the distribution of Birkhoff sums on the scale of $\sqrt{n}$, the local central limit theorem can probe the density at a given point.
Our main result refines the coarse counting of the Prime Orbit Theorem \cite{li2024prime:dirichlet} for expanding Thurston maps.

Let $f \colon S^2 \to S^2$ be an expanding Thurston map and $\phi \colon S^2 \to \mathbb{R}$ be a real-valued H\"{o}lder continuous function.
The topological $2$-sphere $S^2$ is equipped with a \emph{visual metric} $d$ (see Section~\ref{sub:Thurston_maps} for details).
A periodic orbit $\tau = \set[\big]{x, f(x), \dots, f^{n-1}(x)}$ (where $f^n(x) = x$) is called \emph{primitive} if $f^m(x) \ne x$ for each integer $m$ with $1 \leqslant m < n$.
We denote the set of primitive periodic orbits of $f$ by $\mathfrak{P}(f)$.
For each $\tau \in \mathfrak{P}(f)$, we write $\phi(\tau) \coloneqq \sum_{x \in \tau} \phi(x)$.

In this article, we investigate the asymptotic distribution of primitive periodic orbits subject to constraints on their Birkhoff sums.
Specifically, for a given number $\alpha \in \mathbb{R}$ and a sequence $\sequen{I_{n}}$ of intervals contained in a compact set $K \subseteq \mathbb{R}$, we study the asymptotic behavior of
\[
 \pi_{f, \phi}(n; \alpha, I_n) \coloneqq \operatorname{card} \set[\big]{ \tau \in \mathfrak{P}_n(f) : \phi(\tau) - n \alpha \in I_{n}}
\]
as $n \to +\infty$, where $\mathfrak{P}_n(f) \coloneqq \set[\big]{\tau \in \mathfrak{P}(f) : |\tau| = n}$.

To obtain precise estimates, we impose specific conditions on the potential $\phi$ and the sequence $\sequen{I_{n}}$.
We assume that $\phi$ is \emph{eventually positive} and satisfies the \emph{strong non-integrability condition} (see Definitions~\ref{def:eventually positive functions} and~\ref{def:strong non-integrability condition}).
Furthermore, denoting the length of $I_{n}$ by $|I_n|$, we assume that the sequence $\sequen[\big]{|I_{n}|^{-1}}$ exhibits sub-exponential growth, i.e., $\limsup_{n \to +\infty} \frac{1}{n} \log \parentheses[\big]{ |I_{n}|^{-1} } = 0$.

We write $A(n) \sim B(n)$ as $n \to +\infty$ if $\lim_{n \to +\infty} A(n) / B(n) = 1$.

We write $A(n) \sim B(n)$ as $n \to +\infty$ if $\lim_{n \to +\infty} A(n) / B(n) = 1$.

Recall that a postcritically-finite rational map is expanding if and only if it has no periodic critical points (see \cite[Proposition~2.3]{bonk2017expanding}).
Therefore, when we restrict our attention to rational maps, we obtain the following corollary of Theorem~\ref{thm:main theorem} and Remark~\ref{rem:chordal metric visual metric qs equiv}.

\begin{corollary}\label{coro:main theorem for postcritically-finite rational maps}
 Let $f \colon \ccx \mapping \ccx$ be a postcritically-finite rational map without periodic critical points.
 Let $\sigma$ be the chordal metric or the spherical metric on the Riemann sphere $\ccx$, and $\phi \in C^{0, \holderexp} \parentheses[\big]{ \ccx, \sigma }$ be an eventually positive real-valued \holder continuous function with exponent $\holderexp \in (0, 1]$ satisfying the $\holderexp$-strong non-integrability condition (with respect to $f$ and a visual metric).
 Then there exists a unique positive number $\rootpressure > 0$ with topological pressure $P(f, -\rootpressure \phi) = 0$ and there exists $N_f \in \n$ depending only on $f$ such that for each $N \in \n$ with $N \geqslant N_f$, the following statements hold for $F \define f^N$ and $\Phi \define \sum_{i=0}^{N-1} \phi \circ f^i$:

Denote $\alpha \coloneqq \frac{\mathrm{d}}{\mathrm{d} t} P(F, t \Phi) |_{t = -\rootpressure}$ and $\sigma \coloneqq \sqrt{ \frac{\mathrm{d}^2}{\mathrm{d} t^2} P(F, t \Phi) |_{t = -\rootpressure } }$.
 Let $\sequen{I_{n}}$ be a sequence of intervals contained in a compact set $K \subseteq \real$ with $\sequen[\big]{|I_{n}|^{-1}}$ having sub-exponential growth.
 Then
 \[
 \pi_{F, \Phi}(n; \alpha, I_n) \sim \frac{ \int_{I_{n}} e^{\rootpressure t} \, \mathrm{d}t }{ \sqrt{2\pi} \, \sigma } \, \frac{e^{\rootpressure \alpha n}}{n^{3/2}} \qquad \text{as } n \to +\infty.
 \]
\end{corollary}

We now consider the following three quantities:
 \begin{align*}
 A_1(n) &\define \int_{ \abs{t} < \varepsilon \sigma \sqrt{n} } \, \abs[\bigg]{ \frac{\ell_{n}^{-1}}{e^{P(f, -\rootpressure \normpotential)n}} \widehat{\psi}_{n}\parentheses[\Big]{ \frac{t}{2 \pi \sigma \sqrt{n}} } \normpartifun[\Big]{-\rootpressure + \frac{\imaginary t}{\sigma \sqrt{n}}} \, - \, \ell_{n}^{-1} e^{-\frac{t^{2}}{2}} \! \int_{\real} \! \psi_{n}(x) \,\mathrm{d}x } \,\mathrm{d}t, \\
 A_2(n) &\define \int_{ \abs{t} \geqslant \varepsilon \sigma \sqrt{n} } \, \abs[\bigg]{ \frac{\ell_{n}^{-1}}{e^{P(f, -\rootpressure \normpotential)n}} \widehat{\psi}_{n}\parentheses[\Big]{ \frac{t}{2 \pi \sigma \sqrt{n}} } \normpartifun[\Big]{-\rootpressure + \frac{\imaginary t}{\sigma \sqrt{n}}} } \,\mathrm{d}t, \\
 A_3(n) &\define \int_{ \abs{t} \geqslant \varepsilon \sigma \sqrt{n} } \, \abs[\bigg]{ \ell_{n}^{-1} e^{-\frac{t^{2}}{2}} \! \int_{\real} \! \psi_{n}(x) \,\mathrm{d}x } \,\mathrm{d}t.
 \end{align*}
 Here $\varepsilon \in (0, 1)$ is chosen to be smaller than $\min\set[\Big]{ \delta , \frac{\sigma^{2}}{4 C_{\delta}}, t_0 }$, where the constants $\delta$ and $C_{\delta}$ are given by Lemma~\ref{lem:Taylor extension of pressure function in imaginary part}, and the constant $t_0$ is given by Lemma~\ref{lem:partition function estimate:expanding Thurston map:bounded imaginary bounded cases}~\ref{item:lem:partition function estimate:expanding Thurston map:bounded imaginary bounded cases:local part}.
 It follows from Claim~1 that
 \[
 A(n) \leqslant \frac{1}{\sqrt{2\pi}} \parentheses[\big]{ A_1(n) + A_2(n) + A_3(n)} \qquad \text{for each } n \in \n.
 \]
 Thus it suffices to show that $\lim_{n \to +\infty} A_{i}(n) = 0$ for each $i \in \{1, 2, 3\}$.

Let $T > 1$ be the constant given by Proposition~\ref{prop:partition function estimate:expanding Thurston map:unbounded imaginary}.
 By Lemma~\ref{lem:partition function estimate:expanding Thurston map:bounded imaginary bounded cases}~\ref{item:lem:partition function estimate:expanding Thurston map:bounded imaginary bounded cases:intermediate part}, there exists $\vartheta \in (0, 1)$ such that
 \begin{equation} \label{eq:temp:prop:estimate of partition function with periodic orbits:Claim3:medium case}
 \int_{ \varepsilon \leqslant \frac{ \abs{t}}{\sigma \sqrt{n} } \leqslant T } \abs[\bigg]{ \frac{\ell_{n}^{-1}}{e^{P(f, -\rootpressure \normpotential)n}} \widehat{\psi}_{n}\parentheses[\Big]{ \frac{t}{2 \pi \sigma \sqrt{n}} } \normpartifun[\Big]{-\rootpressure + \frac{\imaginary t}{\sigma \sqrt{n}}} } \,\mathrm{d}t
 = \mathcal{O}\parentheses[\big]{ \sigma \sqrt{n} \, \ell_{n}^{-1} \uniformnorm{\widehat{\psi}_{n}} \vartheta^{n} }
 \end{equation}
 as $n \to +\infty$, where $\ell_{n}^{-1} \uniformnorm{\widehat{\psi}_{n}} \leqslant \ell_{n}^{-1} \int_{\real} \! \psi_{n}(x) \,\mathrm{d}x = \int_{\real} \! \psi(y) e^{\rootpressure \ell_{n} y} \,\mathrm{d}y$ is uniformly bounded for $n \in \n$ since $\psi$ has compact support.
 Since $\psi_n \in C^{4}(\real, \real)$, we have that for all $n \in \n$ and $t \in \real$,
 \[
 (2 \pi \imaginary t)^{4} \widehat{\psi}_{n}(t) = \widehat{\psi_{n}^{(4)}}(t).
 \]
 Moreover, since $\psi$ has compact support, we have that
 \[
 \begin{split}
 \ell_{n}^{3} \, \uniformnorm[\Big]{ \widehat{\psi_{n}^{(4)}} }
 \leqslant \ell_{n}^{3} \! \int_{\real} \abs[\big]{ \psi_{n}^{(4)}(x) } \,\mathrm{d}x
 &= \int_{\real} \, \abs[\bigg]{ \sum_{k=0}^{4} \binom{4}{k} \ell_{n}^{k} \, \psi^{(4 - k)}(y) (\rootpressure )^{k} e^{\rootpressure \ell_{n} y} } \,\mathrm{d}y \leqslant C'
 \end{split}
 \]
 for some constant $C' \geqslant 0$, which is independent of $n$.
 Then, by Proposition~\ref{prop:partition function estimate:expanding Thurston map:unbounded imaginary}, there exist $C > 0$ and $\rho \in (0, 1)$ such that for each integer $n \geqslant 2$,
 \begin{align*}
 &\int_{ \abs{t} > T \sigma \sqrt{n} } \, \abs[\bigg]{ \frac{\ell_{n}^{-1}}{e^{P(f, -\rootpressure \normpotential)n}} \widehat{\psi}_{n}\parentheses[\Big]{ \frac{t}{2 \pi \sigma \sqrt{n}} } \normpartifun[\Big]{-\rootpressure + \frac{\imaginary t}{\sigma \sqrt{n}}} } \,\mathrm{d}t \\
 &\qquad \leqslant \int_{ \abs{t} > T \sigma \sqrt{n} } \, \abs[\bigg]{ \ell_{n}^{-1} \widehat{\psi}_{n}\parentheses[\Big]{ \frac{t}{2 \pi \sigma \sqrt{n}} } C \abs[\Big]{ \frac{t}{\sigma\sqrt{n}} }^{2 + \varepsilon} \rho^{n} } \,\mathrm{d}t\\
 &\qquad = \int_{ \abs{t} > T } C \sigma \sqrt{n} \, \ell_{n}^{-1} \rho^{n} \abs{t}^{2 + \varepsilon} \abs[\Big]{ \widehat{\psi}_{n}\parentheses[\Big]{ \frac{t}{2 \pi} } } \,\mathrm{d}t \\
 &\qquad = C \sigma \sqrt{n} \, \ell_{n}^{-4} \rho^{n} \int_{ \abs{t} > T } \abs{t}^{-2 + \varepsilon} \, \ell_{n}^{3} \abs[\Big]{ \widehat{\psi^{(4)}_{n}} \parentheses[\Big]{ \frac{t}{2 \pi} } } \,\mathrm{d}t \\
 &\qquad \leqslant C C' \sigma \sqrt{n} \, \ell_{n}^{-4} \rho^{n} \int_{ \abs{t} > T } \abs{t}^{-2 + \varepsilon} \,\mathrm{d}t.
 \end{align*}
 Combining this with \eqref{eq:temp:prop:estimate of partition function with periodic orbits:Claim3:medium case} and recalling that the sequences $\sequen{\ell_{n}^{-1}}$ are of sub-exponential growth, we establish Claim~3.

Recall that a postcritically-finite rational map is expanding if and only if it has no periodic critical points (see \cite[Proposition~2.3]{bonk2017expanding}).
Therefore, when we restrict our attention to rational maps, we obtain the following corollary of Theorem~\ref{thm:main theorem} and Remark~\ref{rem:chordal metric visual metric qs equiv}.

\begin{corollary}\label{coro:main theorem for postcritically-finite rational maps}
    Let $f \colon \ccx \rightarrow \ccx$ be a postcritically-finite rational map without periodic critical points.
    Let $\sigma$ be the chordal metric or the spherical metric on the Riemann sphere $\ccx$, and $\phi \in C^{0, \beta} \parentheses[\big]{ \ccx, \sigma }$ be an eventually positive real-valued H\"{o}lder  continuous function with exponent $\beta \in (0, 1]$ satisfying the $\beta$-strong non-integrability condition (with respect to $f$ and a visual metric).
    Then there exists a unique positive number $\rootpressure > 0$ with topological pressure $P(f, -\rootpressure \phi) = 0$ and there exists $N_f \in \mathbb{N}$ depending only on $f$ such that for each $N \in \mathbb{N}$ with $N \geqslant N_f$, the following statements hold for $F \coloneqq f^N$ and $\Phi \coloneqq \sum_{i=0}^{N-1} \phi \circ f^i$:

    Denote $\alpha \coloneqq \frac{\mathrm{d}}{\mathrm{d} t} P(F, t \Phi) |_{t = -\rootpressure}$ and $\sigma \coloneqq \sqrt{ \frac{\mathrm{d}^2}{\mathrm{d} t^2} P(F, t \Phi) |_{t = -\rootpressure } }$.
    Let $\sequen{I_{n}}$ be a sequence of intervals contained in a compact set $K \subseteq \mathbb{R}$ with $\sequen[\big]{|I_{n}|^{-1}}$ having sub-exponential growth.
    Then
    \[
        \pi_{F, \Phi}(n; \alpha, I_n) \sim \frac{ \int_{I_{n}} e^{\rootpressure t} \, \mathrm{d}t }{ \sqrt{2\pi} \, \sigma } \, \frac{e^{\rootpressure \alpha n}}{n^{3/2}} \qquad \text{as } n \to +\infty.
    \]
\end{corollary}

\subsection{Strategy and organization}
Our approach relies on a combination of thermodynamic formalism and operator theory, specifically adapted to the branched covering setting.

The main technical obstacle in studying Thurston maps is the presence of critical points, which disrupts the functional analytic properties of the standard Ruelle transfer operator.
To overcome this, we employ the \emph{split Ruelle operators} introduced in \cite{li2024prime:split}.
The idea is to decompose the sphere into ``black'' and ``white'' tiles (based on a checkerboard coloring induced by an invariant Jordan curve) and define a pair of operators acting on functions supported on these tiles.
This construction effectively ``unfolds'' the singularities, allowing us to recover good spectral properties.

To obtain the precise asymptotics required for the local central limit theorem, we need to control the decay of the characteristic function of the Birkhoff sums.
In terms of operator theory, this translates to bounding the spectral radius of the twisted transfer operator $\mathbb{L}_{s\phi}$ as the complex parameter $s$ moves along the imaginary axis.
The detailed estimates are separated into three parts: the unbounded part, the bounded part, and the local part.
For the unbounded part, we employ Dolgopyat-type estimates for the split Ruelle operators established in \cite{li2024prime:split}.
This requires checking a strong non-integrability condition (Definition~\ref{def:strong non-integrability condition}), which in particular implies that the potential is not cohomologous to a constant.
For the bounded part, we employ Ruelle's estimate (see Appendix~\ref{sec:Appendix:Ruelle lemma}).
For the local part, we employ the complex Ruelle--Perron--Frobenius theorem \cite[Theorem~2]{pollicott1984complex} and arguments in \cite{pollicottZetaFunctionsPeriodic1990}.

Finally, to count orbits in intervals $I_n$, we approximate the indicator function $\mathbbm{1}_{I_n}$ by smooth test functions.
We then apply Fourier transforms to relate the smoothed count to partition functions, which allows us to apply the established decay estimates.

\smallskip

The paper is organized as follows.
In Section~\ref{sec:Preliminaries}, we fix our notation, review fundamental concepts from thermodynamic formalism, and recall key results from the theory of expanding Thurston maps.
In Section~\ref{sec:The Assumptions}, we collect the main assumptions used throughout the paper.
The technical core of the paper is Section~\ref{sec:Pressure function and partition function estimates}, where we employ previous results and derive crucial decay estimates for associated partition functions.
These estimates are then used in Section~\ref{sec:Proof of the main theorem} to prove Theorem~\ref{thm:main theorem}.
section/Proof.tex0000664000000000000000000007214415112046574013046 0ustar  rootroot

To prove Theorem~\ref{thm:main theorem}, the approach uses “a combination of thermodynamic formalism and operator theory, specifically adapted to the branched covering setting.” The “main technical obstacle” is that “critical points … disrupt the functional analytic properties of the standard Ruelle transfer operator,” so the argument “employ[s] the \emph{split Ruelle operators}” by “decompos[ing] the sphere into ‘black’ and ‘white’ tiles … and defin[ing] a pair of operators acting on functions supported on these tiles,” which “unfolds the singularities” and restores “good spectral properties.”

For the local central limit theorem asymptotics, one “control[s] the decay of the characteristic function of the Birkhoff sums,” i.e. “bound[s] the spectral radius of the twisted transfer operator $\mathbb{L}_{s\phi}$ as the complex parameter $s$ moves along the imaginary axis.” The needed estimates are split into “three parts: the unbounded part, the bounded part, and the local part”: for the unbounded part use “Dolgopyat-type estimates for the split Ruelle operators,” which “requires checking a strong non-integrability condition … (in particular [implying] that the potential is not cohomologous to a constant)”; for the bounded part use “Ruelle’s estimate”; and for the local part use “the complex Ruelle--Perron--Frobenius theorem … and arguments in \cite{pollicottZetaFunctionsPeriodic1990}.”

Finally, “to count orbits in intervals $I_n$,” one “approximate[s] the indicator function $\mathbbm{1}_{I_n}$ by smooth test functions,” then “apply Fourier transforms to relate the smoothed count to partition functions,” so that the previously established decay estimates can be applied.

To prove the main theorem, the approach uses “a combination of thermodynamic formalism and operator theory, specifically adapted to the branched covering setting.” The “main technical obstacle” is that “critical points … disrupt the functional analytic properties of the standard Ruelle transfer operator,” so the argument “employ[s] the \emph{split Ruelle operators}” by “decompos[ing] the sphere into ‘black’ and ‘white’ tiles … and defin[ing] a pair of operators acting on functions supported on these tiles,” which “unfolds the singularities” and restores “good spectral properties.”

For the local central limit theorem asymptotics, one “control[s] the decay of the characteristic function of the Birkhoff sums,” i.e. “bound[s] the spectral radius of the twisted transfer operator $\mathbb{L}_{s\phi}$ as the complex parameter $s$ moves along the imaginary axis.” The needed estimates are split into “three parts: the unbounded part, the bounded part, and the local part”: for the unbounded part use “Dolgopyat-type estimates for the split Ruelle operators,” which “requires checking a strong non-integrability condition … (in particular [implying] that the potential is not cohomologous to a constant)”; for the bounded part use “Ruelle’s estimate”; and for the local part use “the complex Ruelle--Perron--Frobenius theorem … and arguments in Mark Pollicott, \emph{Zeta functions, periodic points and oscillations of the prime orbit theorem} (1990).”

Finally, “to count orbits in intervals $I_n$,” one “approximate[s] the indicator function $\mathbbm{1}_{I_n}$ by smooth test functions,” then “apply Fourier transforms to relate the smoothed count to partition functions,” so that the previously established decay estimates can be applied.

\label{thm:main theorem}
    Let $f \colon S^2 \to S^2$ be an expanding Thurston map and $d$ be a visual metric on $S^2$ for $f$.
    Let $\beta \in (0, 1]$ and $\phi \in C^{0, \beta}(S^2, d)$ be an eventually positive real-valued H\"{o}lder  continuous function satisfying the $\beta$-strong non-integrability condition (with respect to $f$ and $d$).
    Then there exists a unique positive number $\rootpressure > 0$ with topological pressure $P(f, -\rootpressure \phi) = 0$ and there exists $N_{f} \in \mathbb{N}$ depending only on $f$ such that for each $N \in \mathbb{N}$ with $N \geqslant N_{f}$, the following statement holds for the iterate $F \coloneqq f^{N}$ and the potential $\Phi \coloneqq \sum_{i = 0}^{N - 1} \phi \circ f^{i}$:

    Denote $\alpha \coloneqq \frac{\mathrm{d}}{\mathrm{d} t} P(F, t \Phi) |_{t = -\rootpressure}$ and $\sigma \coloneqq \sqrt{ \frac{\mathrm{d}^2}{\mathrm{d} t^2} P(F, t \Phi) |_{t = -\rootpressure } }$.
    Let $\sequen{I_{n}}$ be a sequence of intervals contained in a compact set $K \subseteq \mathbb{R}$ with $\sequen[\big]{|I_{n}|^{-1}}$ having sub-exponential growth.
    Then
    \[
        \pi_{F, \Phi}(n; \alpha, I_n) \sim \frac{ \int_{I_{n}} e^{\rootpressure t} \, \mathrm{d}t }{ \sqrt{2\pi} \, \sigma } \, \frac{e^{\rootpressure \alpha n}}{n^{3/2}} \qquad \text{as } n \to +\infty.
    \],

Question:
Let $f\colon S^2\to S^2$ be an expanding Thurston map, let $d$ be a visual metric on $S^2$ for $f$, let $\beta\in(0,1]$, and let $\phi\in C^{0,\beta}(S^2,d)$ be an eventually positive real-valued H\"older continuous function satisfying the $\beta$-strong non-integrability condition with respect to $f$ and $d$. For $N\in\mathbb N$, set $F:=f^N$ and $\Phi:=\sum_{i=0}^{N-1}\phi\circ f^i$. If $\tau=\{x,F(x),\dots,F^{n-1}(x)\}$ is a primitive periodic orbit of $F$ (meaning $F^n(x)=x$ and $F^m(x)\neq x$ for $1\le m<n$), write $|\tau|=n$, let $\mathfrak P_n(F)$ be the set of primitive periodic orbits of $F$ of length $n$, and define $\Phi(\tau):=\sum_{y\in\tau}\Phi(y)$. For $\alpha\in\mathbb R$ and intervals $I_n\subseteq K$ with $K\subseteq\mathbb R$ compact, define
\[
\pi_{F,\Phi}(n;\alpha,I_n):=\operatorname{card}\{\tau\in\mathfrak P_n(F):\Phi(\tau)-n\alpha\in I_n\}.
\]
Which statement about existence, uniqueness, and the asymptotic formula for $\pi_{F,\Phi}(n;\alpha,I_n)$ is valid?

There exists a unique positive number $s_0>0$ such that $P(f,-s_0\phi)=0$, and there exists $N_f\in\mathbb N$, depending only on $f$, such that for every $N\ge N_f$, if $F=f^N$ and $\Phi=\sum_{i=0}^{N-1}\phi\circ f^i$, then with
\[
\alpha:=\left.\frac{d}{dt}P(F,t\Phi)\right|_{t=-s_0},\qquad
\sigma:=\sqrt{\left.\frac{d^2}{dt^2}P(F,t\Phi)\right|_{t=-s_0}},
\]
for every sequence of intervals $(I_n)$ contained in a compact set $K\subseteq\mathbb R$ such that $(|I_n|^{-1})$ has sub-exponential growth, one has
\[
\pi_{F,\Phi}(n;\alpha,I_n)\sim \frac{\int_{I_n} e^{s_0 t}\,dt}{\sqrt{2\pi}\,\sigma}\,\frac{e^{s_0\alpha n}}{n^{3/2}}
\qquad\text{as } n\to+\infty.
\]"

Choices:
B.
There exists a unique positive number $s_0>0$ such that $P(f,-s_0\phi)=0$, and for every $N\in\mathbb N$, if $F=f^N$ and $\Phi=\sum_{i=0}^{N-1}\phi\circ f^i$, then with
\[
\alpha:=\left.\frac{d}{dt}P(F,t\Phi)\right|_{t=-s_0},\qquad
\sigma:=\sqrt{\left.\frac{d^2}{dt^2}P(F,t\Phi)\right|_{t=-s_0}},
\]
for every sequence of intervals $(I_n)$ contained in a compact set $K\subseteq\mathbb R$ such that $(|I_n|^{-1})$ has sub-exponential growth, one has
\[
\pi_{F,\Phi}(n;\alpha,I_n)\sim \frac{\int_{I_n} e^{s_0 t}\,dt}{\sqrt{2\pi}\,\sigma}\,\frac{e^{s_0\alpha n}}{n^{3/2}}
\qquad\text{as } n\to+\infty.
\]
C.
There exists a unique positive number $s_0>0$ such that $P(f,-s_0\phi)=0$, and there exists $N_f\in\mathbb N$, depending only on $f$, such that for every $N\ge N_f$, if $F=f^N$ and $\Phi=\sum_{i=0}^{N-1}\phi\circ f^i$, then with
\[
\alpha:=\left.\frac{d}{dt}P(F,t\Phi)\right|_{t=-s_0},
\]
for every sequence of intervals $(I_n)$ contained in a compact set $K\subseteq\mathbb R$ such that $(|I_n|^{-1})$ has sub-exponential growth, one has
\[
\pi_{F,\Phi}(n;\alpha,I_n)=\mathcal O\!\left(\frac{e^{s_0\alpha n}}{n^{3/2}}\right)
\qquad\text{as } n\to+\infty.
\]
D.
There exists a unique positive number $s_0>0$ such that $P(f,-s_0\phi)=0$, and there exists $N_f\in\mathbb N$, depending only on $f$, such that for every $N\ge N_f$, if $F=f^N$ and $\Phi=\sum_{i=0}^{N-1}\phi\circ f^i$, then with
\[
\alpha:=\left.\frac{d}{dt}P(F,t\Phi)\right|_{t=-s_0},\qquad
\sigma:=\sqrt{\left.\frac{d^2}{dt^2}P(F,t\Phi)\right|_{t=-s_0}},
\]
for every sequence of intervals $(I_n)$ contained in a compact set $K\subseteq\mathbb R$, one has
\[
\pi_{F,\Phi}(n;\alpha,I_n)\sim \frac{|I_n|}{\sqrt{2\pi}\,\sigma}\,\frac{e^{s_0\alpha n}}{n^{3/2}}
\qquad\text{as } n\to+\infty.
\]
E.
There exists a unique positive number $s_0>0$ such that $P(f,-s_0\phi)=0$, and there exists $N_f\in\mathbb N$, depending only on $f$, such that for every $N\ge N_f$, if $F=f^N$ and $\Phi=\sum_{i=0}^{N-1}\phi\circ f^i$, then with
\[
\alpha:=\left.\frac{d}{dt}P(F,t\Phi)\right|_{t=-s_0},\qquad
\sigma:=\sqrt{\left.\frac{d^2}{dt^2}P(F,t\Phi)\right|_{t=-s_0}},
\]
for every sequence of intervals $(I_n)$ contained in a compact set $K\subseteq\mathbb R$ such that $(|I_n|^{-1})$ has sub-exponential growth, one has
\[
\pi_{F,\Phi}(n;\alpha,I_n)\sim \frac{\int_{I_n} e^{s_0 t}\,dt}{\sqrt{2\pi}\,\sigma}\,\frac{e^{s_0\alpha n}}{n^{1/2}}
\qquad\text{as } n\to+\infty.
\]

threshold iterate N\ge N_f replaced by all N

dropped the sharp asymptotic constant and equivalence, retaining only a coarse upper bound of the same order

smooth approximation/Fourier-weighted interval term replaced by bare length and sub-exponential hypothesis removed

prime-orbit scale n^{-3/2} replaced by CLT-like n^{-1/2}

The stem itself does not state the conclusion or uniquely reveal the correct option; it only gives the hypotheses and asks which resulting theorem statement is valid.

The item is essentially a theorem-recognition question: under a full list of hypotheses, the student must pick the exact conclusion. It is not a verbatim restatement in the stem, but it remains close to selecting the correct theorem statement.

There is some reasoning pressure because the options differ in subtle but meaningful ways (quantifier on N, sharp asymptotic vs bound, interval factor, power of n). However, it mainly tests precise recall/discrimination rather than constructing a conclusion from first principles.

The distractors are mathematically plausible and target realistic failure modes: overgeneralizing to all N, weakening an asymptotic to an O-bound, dropping the weighted interval term/sub-exponential condition, and confusing n^{-3/2} with n^{-1/2}.

A solid high-level theorem-discrimination MCQ with little stem leakage and strong distractors, but it primarily assesses recognition of a precise result rather than genuinely generative mathematical reasoning.


\label{thmintro:delocalised-trace-identification-simplified}
  Let $\cH$ be an Iwahori Hecke algebra of right-angled type $(W,S)$.  Let $\cO \subseteq W$ be a conjugacy class.  Then there is a unique trace $\vphi_\cO$ on $\cH$ that satisfies $\vphi_\cO(T_w) = \mathbb{1}_\cO(w)$ for every element $w \in W$ that is of minimal length in its conjugacy class.

\begin{remark}
  \label{rem:finite-order-element-cliques}
  Let $\Gamma$ be the commutation graph of a right-angled Coxeter system $(W, S)$.  Denote by $\mathrm{Cliq}(\Gamma)$ the set of cliques in $\Gamma$, that is the set of complete subgraphs of $\Gamma$ (including the empty graph).  Identifying vertices of $\Gamma$ with $S$, we consider the map $C \mapsto \prod_{s \in \rV(C)} s$.  By Remark~\ref{rem:finite-order-element-conjugacy} every finite order element is conjugate to an element in the image of this map.  Further, if $\prod_{s \in \rV(C)} s$ and $\prod_{s \in \rV(D)} s$ are conjugate, they must be equal by Tits solution to the word problem.  It follows that $C \mapsto \prod_{s \in \rV(C)} s$ defines a bijection between $\mathrm{Cliq}(\Gamma)$ and the set of conjugacy classes of finite order elements of $W$.
\end{remark}

\begin{theorem}
  \label{thm:right-angled-cocentre-basis}
  Let $\cH$ be a generic Hecke algebra of right-angled type $(W,S)$ over an algebraically closed field $K$ of characteristic different from $2$. Let $(a_s)_{s\in S}$, $(b_s)_{s\in S}$ be the deformation parameters of $\cH$ and assume that $b_s \neq -\frac{a_s^2}{4}$ for each $s \in S$.  Then $(T_{\cO})_{\cO}$ where $\cO$ runs through the conjugacy classes of $W$ is a basis of the cocentre of $\cH$.
\end{theorem}

\label{sec:introduction}

The study of delocalised traces forms an important tool in noncommutative geometry and index theory, particularly in the analysis of delocalised $\Ltwo$-invariants and higher indices associated with manifolds, discrete groups and their operator algebras. These traces, which extend the familiar notion of the canonical trace on group C*-algebras, allow one to distinguish K-theory classes that are indistinguishable by the standard trace. A foundational instance of their use appears in the work of Lott \cite{lott1999}, who investigated for example delocalised $\Ltwo$-Betti numbers and delocalised $\Ltwo$-torsion of hyperbolic manifolds. The theoretical underpinning for the pairing of delocalised traces with K-theory classes in the context of hyperbolic groups was developed by Puschnigg \cite{puschnigg2010}, who constructed dense subalgebras of group C*-algebras which are at the same time stable under holomorphic functional calculus and host bounded traces that sum coefficients over possibly infinite conjugacy classes. This might be viewed as a natural continuation of the work of Jolissaint \cite{jolissaint1989-k-theory}, who, inspired by the earlier results of Connes, produced natural smooth subalgebras contained in reduced group C*-algebras, remembering the K-theory of their C*-counterparts. More generally, Puschnigg was able to prove that cyclic homology of the group algebra injects into local cyclic homology of the associated reduced group C*-algebra.

While this story is well-developed for group C*-algebras of hyperbolic groups, a parallel and increasingly important direction involves Hecke C*-algebras, which arise as deformations of group algebras associated with Coxeter systems. Historically, spherical and affine Hecke algebras appeared naturally in number theory and representation theory and as such have become a central part of a deep and well-developed theory. They are closely connected to the representation theory of finite groups of Lie type and reductive p-adic groups, respectively. Delorme-Opdam's work in \cite{delormeopdam2008} considered Hecke C*-algebra of affine types and introduced Schwartz algebras in this context. Notably, by Bernstein's theorem, affine Hecke algebras are finitely generated modules over their centre, so that classical methods related the Fourier transform could be employed. These Hecke-Schwartz algebras of affine type were instrumental in Opdam-Solleveld's approach to determine discrete series representations \cite{opdamsolleveld2010-discrete-series} and generalised principal series \cite{delormeopdam2011} of affine Hecke algebras.

Hecke operator algebras of indefinite type drew attention only after the introduction of Hecke von Neumann algebras in the context of weighted $\Ltwo$-cohomology of buildings by Davis-Dymara-Januszkiewicz-Okun \cite{davisdymarajanusykiewiczokun07} around two decades ago. Specifically right-angled Hecke operator algebras since then gained prominence in the operator algebra community due to their rich structural features, which are at the same time amenable to a variety of established methods from the field. Known features of right-angled Hecke operator algebras by now include explicit, non-trivial factor decomposition \cite{garncarek16,raumskalski2023} and absence of Cartan or even strong solidity results \cite{capsers2020,klisse2023-boundaries,borstcasperswasilewski2024} for Hecke-von-Neumann algebras, and the rapid decay property \cite{caspersklisselarsen2021}, simplicity results \cite{caspersklisselarsen2021,klisse2023-simplicity} and K-theory calculations \cite{raumskalski2022} for Hecke C*-algebras. Also compact quantum metric spaces associated with Hecke C*-algebas were very recently introduced and investigated in \cite{klisseperovic25}. Moreover, Hecke operator algebras encode significant aspects of unitary representation theory, particularly through their role in classifying Iwahori-spherical unitary representations of groups acting strongly transitively on buildings such as certain completed Kac-Moody groups. See e.g.\ \cite[Corollary B]{raumskalski2023}.

The cyclic homology of affine Hecke algebras, was clarified in work Baum-Nistor \cite{baumnistor2002} in the purely algebraic setup and by Solleveld \cite{solleveld2009-thesis,solleveld2009-hp} in the operator algebraic setup of Hecke-C*-algebras as well as in the setup of non-commutative geometry provided by Schwartz algebras.

In this work we consider Hecke algebras of right-angled, hyperbolic type. In this setup, delocalised traces account for all of cyclic cohomology, so that a natural first step is to determine traces on such Hecke algebras. This is achieved in the next theorem, which is also proven in the greater generality of generic Hecke algebras in Theorem~\ref{thm:delocalised-trace-identification}. For simplicity we state it in a simplified form for Iwahori-Hecke algebras.

\begin{theorem}
  \label{thm:delocalised-trace-identification}
  Let $\cH$ be a generic Hecke algebra of right-angled type $(W,S)$.  For every conjugacy class $\cO \subseteq W$, the map $\vphi_\cO$ is a trace on $\cH$.  If $(T_{\cU})_{\cU}$ denotes the basis of the cocentre of $\cH$ satisfying $T_{\cU} = T_w + [\cH, \cH]$ for every element of minimal length $w \in \cU$, then $\vphi_\cO(T_\cU) = \delta_{\cO, \cU}$.
\end{theorem}

The next theorem extends Puschnigg's construction of Schwartz algebras for hyperbolic groups to the realm of right-angled Hecke algebras.
\begin{introtheorem}
 \label{introthm:Schwartz-algebra-existence}
 Let $(W,S)$ be a right-angled, hyperbolic Coxeter system and $\cH$ the complex Iwahori-Hecke algebra of type $(W,S)$ with deformation parameter $(q_s)_{s \in S} \in \RR_{> 0}^S$. There is a subalgebra $\cS(W, S, q) \subseteq \Cstarred(W, S, q)$, which
 \begin{itemize}
 \item is dense and closed under holomorphic functional calculus,
 \item and such that for every conjugacy class $\cO \subseteq W$, the trace $\vphi_\cO$ continuously extends to $\cS(W, S, q)$.
 \end{itemize}
\end{introtheorem}

The proof of \cite[Theorem 5.3]{henie2014} applies verbatim to show the following result: the collection $(T_\cO)_{\cO}$ linearly generates the cocentre of any generic Hecke algebra. We will use the following description of the coefficients appearing in linear combinations of these elements.
\begin{lemma}
 \label{lem:non-minimal-element-cocentre}
 Let $\cH$ be a generic Hecke algebra of type $(W,S)$. Let $\cO \subseteq W$ be a conjugacy class and let $w \in \cO$. Then
 \begin{gather*}
 T_w + [\cH, \cH]
 =
 \sum_\cK c_{\cK, w} T_\cK
 \end{gather*}
 for certain coefficients $c_{\cK, w}$, where $\cK$ runs over all conjugacy classes of elements that are Bruhat subordinate to $w$.
\end{lemma}
\begin{proof}
 Let $\ell$ be the minimal length of an element in $\cO$. We prove the claim by induction on $|w| - \ell = k \in \NN_0$. For $k = 0$ there is nothing to prove.

\begin{notation}
 \label{not:delocalised-trace}
 Let $(W, S)$ be a Coxeter system of right-angled type and let $\cH$ be a generic Hecke algebra of type $(W, S)$ defined over $R$. For a conjugacy class $\cO \subseteq W$, we denote by $\Sigma_\cO\colon \cH \lra R$ the sum over all coefficients of minimal length elements in $\cO$, that is
 \begin{gather*}
 \Sigma_\cO(T_w)
 =
 \begin{cases}
 1 & \text{ if $w \in \cO$ and $w$ is of minimal length in $\cO$} \\
 0 & \text{ otherwise}\eqstop
 \end{cases}
 \end{gather*}
 Combining this with Notation~\ref{not:cyclic-reduction-map}, we will write $\vphi_\cO = \Sigma_\cO \circ \Phi$.
\end{notation}

\begin{theorem}
 \label{thm:delocalised-trace-identification}
 Let $\cH$ be a generic Hecke algebra of right-angled type $(W,S)$. For every conjugacy class $\cO \subseteq W$, the map $\vphi_\cO$ is a trace on $\cH$. If $(T_{\cU})_{\cU}$ denotes the basis of the cocentre of $\cH$ satisfying $T_{\cU} = T_w + [\cH, \cH]$ for every element of minimal length $w \in \cU$, then $\vphi_\cO(T_\cU) = \delta_{\cO, \cU}$.
\end{theorem}
\begin{proof}
 In order to show that $\vphi_\cO$ is a trace, it suffices to show that for all $w \in W$ and all $s \in S$ the equality $\vphi_\cO(T_s T_w) = \vphi_\cO(T_w T_s)$ holds. We prove this statement by induction on $|w| \in \NN_0$ and observe that the case $|w| = 0$ is clear. We distinguish several cases.

\begin{theorem}
 \label{thm:delocalised-trace-existence}
 Let $(W,S)$ be a right-angled, hyperbolic Coxeter system and let $\cH$ be the Iwahori-Hecke algebra of type $(W,S)$ with deformation parameter $(q_s)_{s \in S} \in \RR_{> 0}^S$. Let $\cS'(W, S, q), \cS(W, S, q) \subseteq \Cstarred(W, S, q)$ be smooth subalgebras, such that $\cS'(W, S, q)$ is an unconditional completion of $\CC(W, S, q)$ and the natural quasi-derivation
 \begin{gather*}
 \Delta\colon T_w \mapsto \sum_{\substack{w = w_1w_2 \\ \text{ reduced decompositon}}} T_{w_1} \ot T_{w_2}
 \end{gather*}
 continuously extends to a map into the unconditional tensor product $\cS(W, S, q) \lra \cS'(W, S, q) \ot \cS'(W, S, q)$. Then for every conjugacy class $\cO \subseteq W$ the trace $\vphi_\cO$ continuously extends to $\cS(W, S, q)$.
\end{theorem}
\begin{proof}
 Let us first observe that the map $\Sigma_\cO$ summing over the finitely many coefficients of elements of minimal length in $\cO$ extends to a continuous map on $\cS'(W, S, q)$, as $\cS'(W, S, q)$ is an unconditional completion of $\CC(W, S, q)$. Consider maps $\gamma, \rho\colon W \lra W$ defined as in Lemma~\ref{lem:cyclic-reduction-hecke-algebra}. Since $\Delta$ extends continuously to a map $\cS(W, S, q) \lra \cS'(W, S, q) \ot \cS'(W, S, q)$ and since $\cS'(W, S, q)$ is an unconditional completion of $\CC(W, S, q)$, the map $\Delta'\colon \cS(W, S, q) \lra \cS'(W, S, q) \ot \cS'(W, S, q)$ satisfying $\Delta'(T_w) = T_{\gamma(w) \rho(w)} \ot T_{\gamma(w)^{-1}}$ is well-defined and continuous. Let $\Phi\colon \CC(W, S, q) \lra \CC(W, S, q)$ be the unique linear map satisfying $\Phi(T_w) = T_{\gamma(w)^{-1}} T_{\gamma(w)} T_{\rho(w)}$. Then we have $\vphi_\cO = \Sigma_\cO \circ \Phi$ as shown in Theorem~\ref{thm:delocalised-trace-identification}. So a continuous extension of $\vphi_\cO$ to $\cS(W, S, q)$ is obtained by the composition
 \begin{multline*}
 \cS(W, S, q) \stackrel{\Delta'}{\lra}
 \cS'(W, S, q) \ot \cS'(W, S, q) \stackrel{\text{tensor flip}}{\lra} \\
 \cS'(W, S, q) \ot \cS'(W, S, q) \stackrel{\text{multiplication}}{\lra}
 \cS'(W, S, q) \stackrel{\Sigma_\cO}{\lra}
 \CC
 \eqcomma
 \end{multline*}
This finishes the proof.
\end{proof}

Before proving Theorem~\ref{thmintro:pairing}, let us clarify its context and notation beyond what we have done in the introduction already. Recall from \cite[Theorem A.2.1]{bost1990-principe-oka} that the inclusion of a smooth subalgebra into a unital C*-algebra induces an isomorphism in K-theory. We also recall the K-theory calculations for right-angled Hecke C*-algebras from \cite[Section 4]{raumskalski2022}. To this end we use the description of conjugacy classes of finite order elements from Remark~\ref{rem:finite-order-element-cliques}. Denote by $\mathrm{Cliq}(\Gamma)$ the set of cliques in the commutation graph $\Gamma$ of a right-angled Coxeter system $(W, S)$, that is subsets $C \subseteq S$ such that $st = ts$ for all $s,t \in C$. For a clique $C \in \mathrm{Cliq}(\Gamma)$, we write $p_C = \prod_{s \in C} \chi_1^s$. Here $\chi_1^s = \pi_s(\chi_1)$ denotes the image of the projection $\chi_1 \in \CC[\ZZ/2]$ associated with the trivial representation under the linear map extending the group homomorphism $\pi_s\colon \ZZ/2 \lra \CC(W, S, q)^\times$ which satisfies $\pi_s(1) = \frac{2}{q_s^{1/2} + q_s^{-1/2}}T_s + \frac{q_s^{-1/2} - q_s^{1/2}}{q_s^{1/2} + q_s^{-1/2}}$. Then for every deformation parameter $q \in \RR_{> 0}^S$ the map $\mathrm{Cliq}(\Gamma) \lra \rK_0(\Cstarred(W, S, q))\colon C \mapsto [p_C]$ induces an isomorphism $\ZZ^{\mathrm{Cliq}(\Gamma)} \lra \rK_0(\Cstarred(W, S, q))$.
\begin{proof}[Proof of Theorem~\ref{thmintro:pairing}]
 As described above, for $D \in \mathrm{Cliq}(\Gamma)$ we have $p_D = \prod_{s \in D} \chi_1^s$ where $\chi_1^s$ is the projection associated with the trivial representation of the unitary representation $\ZZ/2 \mapsto \CC(W, S, q)$ mapping $1$ to $\frac{2}{q_s^{1/2} + q_s^{-1/2}}T_s + \frac{q_s^{-1/2} - q_s^{1/2}}{q_s^{1/2} + q_s^{-1/2}}$. We have $\chi_1^s = \mathbb{1}_{(0,\infty)}(T_s) = \frac{1}{q_s^{1/2} + q_s^{-1/2}} (T_s + q_s^{-1/2}) =\frac{1}{q_s^{1/2} + q_s^{-1/2}} T_s + \frac{1}{1 + q_s}$. Writing $T_D = \prod_{s \in D} T_s$, we find that
 \begin{gather*}
 p_D = \sum_{C \subseteq D} \bigl ( \prod_{s \in C} \frac{1}{q_s^{1/2} + q_s^{-1/2}} \bigr ) \bigl ( \prod_{s \in D \setminus C} \frac{1}{1 + q_s} \bigr ) T_C
 \eqstop
 \end{gather*}
 Since all elements in the support of $p_D$ have minimal length and are pairwise non-conjugate, we find that
 \begin{gather*}
 \vphi_C(p_D)
 =
 \Sigma_C(p_D)
 =
 \begin{cases}
 0 & C \nsubseteq D \eqcomma \\
 \prod_{s \in C} \frac{1}{q_s^{1/2} + q_s^{-1/2}} \prod_{s \in D \setminus C} \frac{1}{1 + q_s} & C \subseteq D \eqstop
 \end{cases}
 \end{gather*}
 Now the claimed formula for $\vphi_C(p_D)$ follows.

We mention that in order to prove this result, we establish in Theorem~\ref{thm:right-angled-cocentre-basis} that the cocentre of Iwahori Hecke algebra of right-angled type has a basis indexed by representatives of conjugacy classes.  The problem of describing the cocentre of Hecke algebras is well-known in the context of character theory.  See e.g. \cite{geckpfeiffer2000-characters} for spherical Coxeter types and \cite{henie2014} for affine Coxeter types.  The recent work \cite{chen2025-centralizers} explicitly states folklore conjectures about the cocentre for indefinite Coxeter types.

The next theorem extends Puschnigg's construction of Schwartz algebras for hyperbolic groups to the realm of right-angled Hecke algebras.
\begin{introtheorem}
  \label{introthm:Schwartz-algebra-existence}
  Let $(W,S)$ be a right-angled, hyperbolic Coxeter system and $\cH$ the complex Iwahori-Hecke algebra of type $(W,S)$ with deformation parameter $(q_s)_{s \in S} \in \RR_{> 0}^S$.  There is a  subalgebra $\cS(W, S, q) \subseteq \Cstarred(W, S, q)$, which
  \begin{itemize}
  \item is dense and closed under holomorphic functional calculus,
  \item and such that for every conjugacy class $\cO \subseteq W$, the trace $\vphi_\cO$ continuously extends to $\cS(W, S, q)$.
  \end{itemize}
\end{introtheorem}

From the point of view of operator algebras, one of the main purposes of delocalised traces is to determine K-theory classes via their pairing.  The K-theory of right-angled Hecke C*-algebras was calculated in \cite{raumskalski2022}.  We briefly recall that for every clique $C$ in the commutation graph of a right-angled Coxeter system there is a projection $p_C$, such that the classes $([p_C])_C$ freely generate K-theory.  In Remark~\ref{rem:finite-order-element-cliques}, we observe that these cliques are in natural bijection with finite order elements of a right-angled Coxeter system.  We obtain the following result, which determines the pairing of K-theory and delocalised traces.
\begin{introtheorem}
  \label{thmintro:pairing}
  Let $(W,S)$ be a right-angled, hyperbolic Coxeter system and $(q_s)_{s \in S} \in \RR_{> 0}^S$ a deformation parameter.  Denote by $\mathrm{Cliq}(\Gamma)$ the set of cliques of the commutation graph $\Gamma$ of $(W, S)$, identified with the set of conjugacy classes of finite order elements in $W$.  For a clique $C \in \mathrm{Cliq}(\Gamma)$ let $p_C$ be the associated projection and let $\vphi_C$ be the delocalised trace associated with $C$.  Then the pairing between K-theory of $\Cstarred(W, S, q)$ and delocalised traces of finite order elements is determined by the formula
  \begin{gather*}
    \vphi_C(p_D) =
    \begin{cases}
      0 & C \nsubseteq D\eqcomma \\
      \prod_{s \in C} \frac{1}{q_s^{1/2} + q_s^{-1/2}} \prod_{s \in D \setminus C} \frac{1}{1 + q_s} & C \subseteq D \eqstop
    \end{cases}
  \end{gather*}
  In particular, the pairing $\rK_0(\Cstarred(W,S,q)) \times \RR^{\mathrm{Cliq(\Gamma)}} \lra \RR$ is non-degenerate.
\end{introtheorem}

As mentioned above, in the last section of \cite{puschnigg2010} Puschnigg used his main results to obtain certain decompositions of cyclic cohomology for his smooth algebras and the group C*-algebras of hyperbolic groups, close in spirit to the computation of cyclic homology of group rings due to Burghelea \cite{burghelea1985, burghelea2023-erratum-for-1985}. This line of research was later continued, using the Jolissaint's rapid decay subalgebras, for example in \cite{jiogleramsey14} and in \cite{engel20}. Our results in principle open the way to obtaining similar decompositions of cyclic cohomology in the context of Hecke algebras, such as the Fr{\'e}chet algebra $\cS(W, S, q)$ appearing in Theorem \ref{introthm:Schwartz-algebra-existence}. However, the setting of right-angled hyperbolic Coxeter types remains too narrow to make such an endeavour worthwhile at present, since higher cyclic cohomology is expected to vanish.  Further work in this direction, would chime with the recent progress in understanding noncommutative geometric properties of the Iwahori-Hecke algebras such as in the work \cite{klisseperovic25} already mentioned.

This paper is organised as follows: after this introduction, in the preliminaries we describe background material and fix notation, focusing on Coxeter groups and Hecke algebras.  Then in the third section, we investigate the combinatorics of the multiplication in right-angled Hecke algebras, which plays a major role in this work.  In the fourth section, we adapt work of Blackadar-Cuntz and Puschnigg on differential seminorms and quasi-derivations to the needed generality.  In the fifth section, we use this framework in order to exhibit specific smooth subalgebras of the Hecke algebras under consideration, to which the natural quasi-derivation extends.  In the sixth section, we describe traces on right-angled Hecke algebras, and in that last section combine this with the previous results on smooth subalgebras, to obtain delocalised traces on Hecke-Schwartz algebras and calculate their pairing with K-theory.

\subsection*{Acknowledgements}

SP was partially supported by European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement no. 677120-INDEX) and the Knut and Alice Wallenberg Foundation (grant number KAW 2020.0252).

SR was partially supported by the Swedish Research Council (grant number 2018-04243) and the German Research Foundation (DFG project no. 550184791).

PN was supported by the National Science Center Grant Maestro-13 UMO-2021/42/A/ST1/00306.

The authors would like to thank IMPAN and its B{\c e}dlewo Conference Center as well as the University to Potsdam for their hospitality during respective visits.

To prove Theorem~\ref{thmintro:delocalised-trace-identification-simplified}, the introduction says that the authors first establish in Theorem~\ref{thm:right-angled-cocentre-basis} that “the cocentre of Iwahori Hecke algebra of right-angled type has a basis indexed by representatives of conjugacy classes.”

Question: Let \(\mathcal H\) be an Iwahori Hecke algebra of right-angled type associated with a Coxeter system \((W,S)\), and let \(\{T_w\}_{w\in W}\) denote its standard basis. Fix a conjugacy class \(\mathcal O\subseteq W\). A trace on \(\mathcal H\) means a linear functional \(\varphi:\mathcal H\to\mathbb C\) such that \(\varphi(xy)=\varphi(yx)\) for all \(x,y\in\mathcal H\). Which statement holds about traces determined by \(\mathcal O\)?

There exists a unique trace \(\varphi_{\mathcal O}:\mathcal H\to\mathbb C\) such that for every \(w\in W\) whose length is minimal in its conjugacy class, \(\varphi_{\mathcal O}(T_w)=\mathbb 1_{\mathcal O}(w)\), where \(\mathbb 1_{\mathcal O}(w)=1\) if \(w\in\mathcal O\) and \(0\) otherwise.

Choices:
B. There exists a unique trace \(\varphi_{\mathcal O}:\mathcal H\to\mathbb C\) such that for every \(w\in W\), \(\varphi_{\mathcal O}(T_w)=\mathbb 1_{\mathcal O}(w)\), where \(\mathbb 1_{\mathcal O}(w)=1\) if \(w\in\mathcal O\) and \(0\) otherwise.
C. There exists a trace \(\varphi_{\mathcal O}:\mathcal H\to\mathbb C\) such that for every \(w\in W\) whose length is minimal in its conjugacy class, \(\varphi_{\mathcal O}(T_w)=\mathbb 1_{\mathcal O}(w)\), where \(\mathbb 1_{\mathcal O}(w)=1\) if \(w\in\mathcal O\) and \(0\) otherwise.
D. For every conjugacy class \(\mathcal O\subseteq W\), there exists a unique linear functional \(\varphi_{\mathcal O}:\mathcal H\to\mathbb C\) such that for every \(w\in W\) whose length is minimal in its conjugacy class, \(\varphi_{\mathcal O}(T_w)=\mathbb 1_{\mathcal O}(w)\); moreover, any such linear functional is automatically a trace.
E. There exists a unique trace \(\varphi_{\mathcal O}:\mathcal H\to\mathbb C\) such that for every \(w\in W\) of minimal length in \(W\), \(\varphi_{\mathcal O}(T_w)=\mathbb 1_{\mathcal O}(w)\), where \(\mathbb 1_{\mathcal O}(w)=1\) if \(w\in\mathcal O\) and \(0\) otherwise.

The stem does not explicitly state the correct theorem or uniquely signal option A. It gives background definitions and asks which statement is true, without directly leaking the answer.

The correct choice is essentially the theorem statement itself: existence and uniqueness of a trace determined by values on minimal-length representatives in a conjugacy class. This makes the item close to a direct restatement rather than a nontrivial application.

Some reasoning is needed to distinguish uniqueness vs. mere existence and the role of minimal-length representatives, but the item mainly tests precise recall/discrimination of a known result rather than generative mathematical reasoning.

The distractors are plausible and mathematically targeted: overstrengthening to all w, weakening by dropping uniqueness, replacing trace by linear functional, and confusing classwise minimal length with global minimal length. These reflect realistic failure modes.

A well-constructed theorem-discrimination MCQ with strong distractors and no major answer leakage, but it is largely a direct recall/restatement item rather than one that strongly tests generative reasoning.