\label{theorem: radius of essential spectrum}
    On the half-plane $\Re(\beta) >(1-\alpha)/2$, the radius $\rho_e(L_\beta):= \sup_{\lambda\in \sigma_\varepsilon(L_\beta)}\left\lvert\lambda\right\rvert$ of the essential spectrum satisfies
    \[\rho_e(L_\beta)\leq \rho(\mathcal{L}_{\Re(\beta) +\alpha}):=\sup_{\lambda\in \sigma(\mathcal{L}_{\beta+\alpha})}\left\lvert\lambda\right\rvert .\]
    Furthermore, the generalised eigenspaces belonging to eigenvalues $\lambda$ with $\left\lvert\lambda\right\rvert>\rho_e(L_\beta)$ consist of analytic functions which extend to generalised eigenfunctions of $\mathcal{L}_\beta$.

\begin{theorem}\label{theorem: radius of essential spectrum}
    On the half-plane $\Re(\beta) >(1-\alpha)/2$, the radius $\rho_e(L_\beta):= \sup_{\lambda\in \sigma_\varepsilon(L_\beta)}\abs{\lambda}$ of the essential spectrum satisfies
    \[\rho_e(L_\beta)\leq \rho(\mathcal{L}_{\Re(\beta) +\alpha}):=\sup_{\lambda\in \sigma(\mathcal{L}_{\beta+\alpha})}\abs{\lambda} .\]
    Furthermore, the generalised eigenspaces belonging to eigenvalues $\lambda$ with $\abs{\lambda}>\rho_e(L_\beta)$ consist of analytic functions which extend to generalised eigenfunctions of $\mathcal{L}_\beta$.
\end{theorem}

\begin{remark}\label{remark: abstract}
    The zeroes of $Z(s)$ on the line $s=1/2+it$ are those for which there are eigenvalues of $\Delta$ called \enquote{Maass cusp forms} with eigenvalue $s(1-s)$. Furthermore the multiplicity of the zeroes and the eigenspaces of $\Delta$ match. The other zeroes of $Z(s)$ are a simple zero at $s=1$ and at the values of $s$ for which $2s$ is a nontrivial zero of the Riemann zeta function $\zeta$. Since $\rho(\mathcal{L}_a)<1$ when $a>1$, we see that the eigenspaces of $\mathcal{L}_\beta$ and $L_\beta$ for the eigenvalues $\lambda=\pm 1$ are identical when $1>\alpha>1-\Re(\beta)$. In particular, the eigenspaces associated to the Maass cusp forms are preserved when $\alpha >1/2$, and the eigenspaces associated to the nontrivial zeroes of the Riemann zeta function\footnote{In fact, the zeroes of $Z(s)$ associated with the nontrivial zeroes of $\zeta$ always occur due to $1$ being an eigenvalue of $\mathcal{L}_\beta$.} to the right of the critical line are preserved when $\alpha=3/4$.
\end{remark}

Let $G:(0,1)\to [0,1)$ be the Gauss map
\begin{equation}\label{equation: gauss map}
 G(x) = \frac{1}{z}-\left\lfloor\frac{1}{z}\right\rfloor.
\end{equation}
We associate to $G$ the following family of transfer operators which we formally define as acting on some space of functions $f:[0,1]\to \mathbb{C}$:
\begin{equation}\label{equation: Mayer transfer operator}
 \mathcal{L}_{\beta}(f)(z)=\sum_{n=1}^\infty \frac{1}{(n+z)^{2\beta}}f\left(\frac{1}{n+z}\right),
\end{equation}
where $\beta \in \{z\in \mathbb{C}: \Re(\beta)>1/2\}$. When $\beta=1$, this is the familiar Ruelle-Perron-Frobenius operator for $G$ and can formally be interpreted as the right-adjoint of the Koopman operator $f\mapsto f\circ G$. The operators in the case $\beta \neq 1$ were first introduced by Mayer \cite{Mayer90}, who considered them acting on the Banach space of functions $A_\infty(D)$ which are holomorphic on $D:=\{z\in\mathbb{C}: \left\lvertz-1\right\rvert< 3/2\}$ with continuous extension to $\overline{D}$.

For this choice of Banach space, Mayer also showed that the operator-valued function $\beta\mapsto \mathcal{L}_{\beta}$ admits a meromorphic continuation on $\mathbb{C}$, with simple poles at $\beta=k/2$ for $k\in\{1,0,-1,-2,\ldots\}$. The continuation for $\Re(\beta) >0$ can be obtained by rewriting $f=f(0)+f^\star$ with $f^\star:=f-f(0)$ and noting that
\begin{equation}\label{equation: Mayer transfer extended}
 \mathcal{L}_{\beta}(f)(z)=f(0)\zeta_H(2\beta,z+1)+\sum_{n=1}^\infty \frac{1}{(n+z)^{2\beta}}(f^\star)\left(\frac{1}{n+z}\right),
\end{equation}
where $\zeta_H(s,z)=\sum_{n=0}^\infty (n+z)^{-s}$ is the Hurwitz zeta function, which has an analytic continuation for all $s\neq 1$ and a simple pole of residue $1$ at $s=1$. In fact, it can further be analytically continued to $\beta \geq -k/2$ for any $k\in\mathbb{N}$ by noticing that
\begin{equation}\label{equation: Mayer transfer extended2}
 \mathcal{L}_{\beta}(f)(z)=\sum_{n=0}^k\frac{d^nf}{dz^n}(0)\frac{\zeta_H(n+2\beta,z+1)}{n!}+\sum_{n=1}^\infty \frac{1}{(n+z)^{2\beta}}(f^\star)\left(\frac{1}{n+z}\right),
\end{equation}
where $f^\star(z)=f(z)- \sum_{n=0}^k\frac{d^nf}{d^nz}(0)z^n/n!$.
Furthermore $\mathcal{L}_{\beta}$ is a nuclear operator on $A_\infty(D)$ of order $0$ for all $\beta$.

The Mayer transfer operator encodes geometric information about the modular surface, which is the quotient orbifold $\mathbf{M}$ obtained from considering the Poincaré upper half plane $\mathbb{H}:\{x+iy:x,y\in \mathbb{R}, y>0\}$ equipped with the usual Riemannian metric $ds^2=(dx^2+dy^2)/y^2$ and identifying points via equivalence relation $\mathbb{H}\ni z\sim (az+b)/(cz+d)$ for all $a,b,c,d\in\mathbb{Z}$ with $ad-bc=1$. Since $\mathcal{L}_{\beta}$ is a nuclear operator, the Fredholm determinants $\det(1-\mathcal{L}_{\beta})$ and $\det(1+\mathcal{L}_{\beta})$ are well-defined. These relate to Selberg Zeta function $Z(s)$ by the formula
\begin{equation}\label{equation: zeta functions}
 Z(s)= \det(1-\mathcal{L}_s)(1+\mathcal{L}_s),
\end{equation}
see \cite{Mayer91}.
The Selberg zeta function is closely connected with the spectral theory of the Laplace-Beltrami operator $\Delta:=-y^2(d^2/dx^2+d^2/dy^2)$ \cite{Iwaniec}. Equation \eqref{equation: Mayer transfer extended} is thus a way of connecting the spectral theory of $\Delta$ with that of $\mathcal{L}_\beta$. This connection has been elaborated upon by Lewis, Zagier, Mayer and Chang \cite{Lewis97,ZagierLewis,MayerChang96,MayerChang98}.

From a dynamical perspective, it is interesting to see to what extent the aforementioned spectral properties of $\mathcal{L}_s$ survive when we consider e.g. Hölder continuous functions instead. For $\alpha\in (0,1)$, let $C^\alpha([0,1])$ denote the space of $\alpha$-Hölder functions on $[0,1]$. If $\alpha = k+\eta$ with $k\in\mathbb{N}$ and $\eta\in [0,1)$, let $C^\alpha([0,1])$ be the set of functions for which the $k$-th derivative exists and is continuous or $\eta$-Hölder if $\eta=0$ or $\eta\in (0,1)$ respectively. Denote by $L_\beta$ the operator \eqref{equation: Mayer transfer extended2} acting on $C^\alpha([0,1])$, which we shall show to be well-defined and continuous for large enough $\alpha$. These operators are not compact, but we can decompose the spectrum $\sigma(L_\beta)$ into the essential spectrum $\sigma_e(L_\beta)$ and the discrete spectrum $\sigma_{\mathrm{disc}}(L_\beta)$ which consists of isolated eigenvalues of finite algebraic multiplicity. We prove the following.

\begin{equation}\label{equation: Mayer transfer extended}
    \mathcal{L}_{\beta}(f)(z)=f(0)\zeta_H(2\beta,z+1)+\sum_{n=1}^\infty \frac{1}{(n+z)^{2\beta}}(f^\star)\left(\frac{1}{n+z}\right),
\end{equation}

\begin{equation}\label{equation: Mayer transfer extended2}
    \mathcal{L}_{\beta}(f)(z)=\sum_{n=0}^k\frac{d^nf}{dz^n}(0)\frac{\zeta_H(n+2\beta,z+1)}{n!}+\sum_{n=1}^\infty \frac{1}{(n+z)^{2\beta}}(f^\star)\left(\frac{1}{n+z}\right),
\end{equation}

Many of the usual theorems in complex analysis continue to hold for Banach or topological vector space-valued functions. In particular, holomorphic implies infinitely differentiable with a locally converging Taylor expansion. The unique extension theorem also continues to hold.
Finally, note that it is straightforward to show that $\beta\to L_\beta$ is a meromorphic function on $\Re(\beta)>(1-\alpha)/2$ for any $\alpha\in\mathbb{R}_{>0}$ with derivative
\[ \left(\frac{d L_\beta}{d\beta}\right)(f)(z)= -\sum_{n=0}^k(n+2\beta)\frac{d^nf}{dz^n}(0)\frac{\zeta_H(n+2\beta+1,z+1)}{n!} -2 \sum_{n=1}^\infty \frac{\ln(n+z)}{(n+z)^{2\beta}}(f^\star)\left(\frac{1}{n+z}\right).\]
\section{Uniformly Bounded Essential Spectrum}\label{section: uniformly bounded essential spectrum}
By \eqref{equation: Nussbaum}, we can estimate the essential spectral radius by finding for each $l\in \mathbb{N}$ a compact operator $T'$ for which $\|L_\beta^l - T' \|$ is small. A first naive approach would be to consider the operator $P$ which sends $f$ to a piecwise linear approximation that interpolates $f$ on the points $0,\frac{1}{N},\frac{2}{N},\cdots,1$ and considering $\|L_\beta^l-L_\beta^l\circ P\|$ as $N\to \infty$. For $\Re(\beta)>0$, we indeed have by Lemma \ref{lemma: mayer operator inequalities} that for any $\varepsilon>0$ there exists some $N$ for which $\lvert (L_\beta-L_\beta\circ P) (f)\rvert_\alpha(x)\leq \mathfrak{L}_{\Re(\beta)+\alpha}((1+\varepsilon)|f-P(f)|_\alpha)(x)+\varepsilon$. However, we do not have sufficient control over the $\|\cdot\|_\infty$-norm. We remedy this by considering a \textit{countable} interpolation which ensures $(L_\beta^l\circ P)f$ remains a (nonlinear) interpolation of
$L_\beta^l f$.

Let us now prove most of Theorem \ref{theorem: radius of essential spectrum}.
\begin{theorem}\label{theorem: spectral theorem}
Let $0<\alpha<1 $. Then for all $\beta$ with $\Re(\beta)>(1-\alpha)/2$ the transfer operator $L_\beta$ satisfies
\[\rho_e(L_\beta) \leq \rho(\mathfrak{L}_{\Re(\beta)+\alpha}).\]
\end{theorem}
\begin{proof}
 By Nussbaum's formula and Proposition \ref{proposition: compact operator}, it suffices to prove that there is some constant $C''>0$ for which \[\limsup_{N\to\infty} \|L_{\beta}^l -L_{\beta}^l \circ P_{l,N} \| \leq C''\left\|\mathfrak{L}_{\Re(\beta)+\alpha}^l\right\| \text{ for all } l.\]

\end{proof}
\begin{remark}\label{remark: spectral radius}
By using the spectral radius formula $\rho(\mathfrak{L}_{\Re(\beta)+\alpha})=\limsup_{l\to\infty}\|\mathfrak{L}_{\Re(\beta)+\alpha}^l\|^{1/l}$ and bounding every function $f\in L^\infty([0,1])$ by a constant, we obtain that $\rho(\mathfrak{L}_{\Re(\beta)+\alpha})\leq \rho(\mathcal{L}_{\Re(\beta)+\alpha})$ and hence that
\[\rho(\mathfrak{L}_{\Re(\beta)+\alpha}) = \lambda_1(\Re(\beta)+\alpha),\]
where $\lambda_1(t)$ for $t>1/2$ is the eigenvalue of maximum modulus of $\mathcal{L}_{t}$. The eigenvalue is guaranteed to be simple and positive by the Perron-Frobenius theorem. In fact $\lambda_1(t)$ is a real-analytic and decreasing function of $t$ with $\lambda(1)=1$, see \cite{Mayer90,Vallee98}.
\end{remark}
Theorem \ref{theorem: radius of essential spectrum} for $\alpha \in (0,1)$ then follows from Remark \ref{remark: spectral radius} and from the following corollary of Theorem \ref{theorem: spectral theorem}.
\begin{corollary}\label{Corollary: Ruelle}
 Let $\lambda \in \mathbb{C}$ satisfy $\abs{\lambda}>\rho(\mathfrak{L}_{\Re(\beta)+\alpha})$. Then $\lambda$ is an eigenvalue of $L_\beta$ if and only if it is an eigenvalue of $\mathcal{L}_\beta$. In this case the respective eigenspaces $E_\lambda^\alpha$ and $E_\lambda^\omega$ are equal.
\end{corollary}
This is an example of a more general principle for operators on Banach spaces, see e.g. \cite{Grabiner}. For our purposes, we may directly adapt a proof due to Ruelle (Corollary 3.3 in \cite{Ruelle89}), which we provide in the appendix.

\end{proof}
We see by Proposition $4.1$ that a generic solution of \eqref{equation: three term functional} in $C^\alpha((-1,\infty))$ satisfies $f= O(z^{1-2\Re(\beta)})$ if $\Re(\beta)>(1-\alpha)/2$. On the other hand, we see that the condition $Q=0$ in \eqref{equation: Qinfinity} or in \eqref{equation: Q0} is equivalent to the condition $L_\beta f= \lambda f$, which by Theorem \ref{theorem: radius of essential spectrum} implies $f$ extends to a holomorphic function in $\mathbb{C}\setminus (-\infty,-1]$ if $\abs{\lambda}\leq \rho(\mathcal{L}_{\Re(\beta)+\alpha})$. For example, we have the following corollary of Proposition \ref{equation: proposition: }, which we can view as a stronger version of the Bootstrapping procedure by Lewis and Zagier in \cite{ZagierLewis}.
\begin{corollary}
 Let $\Re(\beta)=1/2$. Suppose $f:(-1,\infty)\to \mathbb{C}$ is some solution of \eqref{equation: three term functional} for $\abs{\lambda} = 1$. Assume furthermore it satisfies the mild growth condition $f(z)= C_0x^{1-2\beta}+o(1)$ as $z\to \infty$.
 Then if $f$ is $1/2+\varepsilon$-Hölder for some $\varepsilon>0$, it is automatically holomorphically extendable to the cut plane $\mathbb{C}\setminus (-\infty,-1]$.
\end{corollary}
This corollary immediately follows from Theorem \ref{theorem: radius of essential spectrum} and the fact that $\rho(\mathcal{L}_a)<1$ if $a>1$. We remark that when $\Re(\beta)=1/2$ and $\lambda =1$ or $-1$, the space of eigenvalues of $\mathcal{L}_{\beta}$ is in bijection with the space of even, respectively uneven Maass cusp forms on the modular surface with eigenvalue $\beta(1-\beta)$ and we have that $C_0=0$, i.e. $f(z)= o(1)$ as $z\to \infty$, see \cite{Lewis97, ZagierLewis}.

\begin{proposition}\label{Proposition: Constructing Q}
 Let $\alpha>0$, $\lambda\in\mathbb{C}\setminus\{0\}$ and let $Q\in C^\alpha(\mathbb{R}/\mathbb{Z})$.
 For all $\beta$ for which $\Re(\beta)> (1-\alpha)/2$ and \[r(\mathcal{L}_{\Re(\beta)+\alpha})< \abs{\lambda},\]
 the function
 \[f=(1 - \lambda^{-1}L_\beta)^{-1}Q,\]
 is a meromorphic expression over all $\beta$ satisfying the conditions in this proposition. Furthermore $f$ satisfies \eqref{equation: three term functional}, \eqref{equation: Qinfinity} and \eqref{equation: Q0}.
\end{proposition}
\begin{proof}
 We interpret $(\lambda - L_\beta)^{-1}$ in the sense of Remark \ref{remark: extending L_beta}. More specifically, for $L_\beta$ acting on $ C^r([a,(1-a)^{-1}])$, the expression $(1 - \lambda^{-1}L_\beta)^{-1}\left(\rstr{Q}{[a,(1-a)^{-1}]}\right)$ is meromorphic and satisfies
\begin{equation}\label{equation: expansion for resolvent}
 \begin{split}
 &(1 - \lambda^{-1}L_\beta)^{-1}\left(\rstr{Q}{[a,(1-a)^{-1}]}\right)(z)=\sum_{n=0}^\infty \lambda^{-n}L_\beta^n\left(\rstr{Q}{[a,(1-a)^{-1}]}\right)(z).\\
 &= Q(z)+\sum_{n=1}^\infty \lambda^{-n}\sum_{a_1,\ldots,a_n\in\mathbb{N}} Q([a_1,\ldots,a_n+z])\Pi_{k=1}^n[a_k,\ldots,a_n+z]^{2\beta}
 \end{split}
\end{equation}
if $\Re(\beta)$ is large enough. By Theorem \ref{theorem: radius of essential spectrum}, we see that $(1 - \lambda^{-1}L_\beta)^{-1}$ is a meromorphic operator-valued function for all $a\leq1$. Hence the functions $(1 - \lambda^{-1}L_\beta)^{-1}\left(\rstr{Q}{[a,(1-a)^{-1}]}\right)$ are
meromorphic functions over the half-plane consisting of all $\beta$ satisfying the conditions in the proposition. We see by the second line of \eqref{equation: expansion for resolvent} and uniqueness of analytic continuation that these functions agree for different $a$ on the intersection of their domains. Hence we may define $(1 - \lambda^{-1}L_\beta)^{-1}Q(z)$ for any $z\in (-1,\infty)$ by choosing $a$ in \eqref{equation: expansion for resolvent} small enough.

The statement in the abstract concerning Maass cusp forms and nontrivial zeroes of the Riemann zeta function, which we make more precise in Remark \ref{remark: abstract}, turns out to follow immediately from this theorem.

Comparing the behaviour of transfer operators for different levels of regularity is a somewhat common theme in dynamics. One typically expects for a transfer operator associated to a sufficiently \enquote{nice} expanding map that the essential spectral radius vanishes for the operator acting on spaces of increasingly smooth functions, see e.g. \cite{Butterley}. In a sense the above theorem is therefore not too suprising. Indeed, we shall show that if $\alpha$ is a natural number, the proof of Theorem \ref{theorem: radius of essential spectrum} is a mundane application of standard techniques.

 However, the proof of the case $1/2<\alpha < 1$ is perhaps more interesting. In that case, we still obtain uniformly bounded estimates for $\rho_e(L_\beta)$ on vertical lines $\Re(\beta)=\sigma$ even when $\sigma < 1/2$, where the function $\beta\mapsto L_\beta$ is unbounded. However, by \eqref{equation: Mayer transfer operator}, the unboundedness in $\beta$ is due to the contribution rank one operator $f\mapsto f(0)\zeta_H(2\beta,z+1)$, which does not contribute to the essential spectral radius. Nevertheless, we do not obtain this result using a Doeblin-Fortet-Lasota-Yorke inequality, so our proof is substantially different from the standard approaches to these type of problems.

In Section \ref{section: three-term functional equation}, we interpret this result in terms of Lewis' three-term functional equation. In particular, we can interpret this as an extension of the \enquote{bootstrapping} result of Lewis and Zagier, who proved in \cite{ZagierLewis} that real-analytic solutions of this equation on $(1,\infty)$ satisfying certain growth conditions automatically extend to holomorphic functions on the cut plane $\mathbb{C}\backslash(-\infty,-1]$. We show that for $\Re(\beta)>0$, we may even replace real-analyticity with a \textit{Hölder condition} and obtain that it is still the restriction of a holomorphic function on $\mathbb{C}\backslash(-\infty,-1]$.

If $\alpha$ is a natural number, “the proof of Theorem \ref{theorem: radius of essential spectrum} is a mundane application of standard techniques.”

For $1/2<\alpha<1$, the proof is “substantially different from the standard approaches”: one “still obtain[s] uniformly bounded estimates for $\rho_e(L_\beta)$ on vertical lines $\Re(\beta)=\sigma$ even when $\sigma<1/2$, where the function $\beta\mapsto L_\beta$ is unbounded,” and this is because “the unboundedness in $\beta$ is due to the contribution rank one operator $f\mapsto f(0)\zeta_H(2\beta,z+1)$, which does not contribute to the essential spectral radius.” The result is “not obtain[ed]… using a Doeblin-Fortet-Lasota-Yorke inequality.”

If $\alpha$ is a natural number, the proof of the main theorem is a mundane application of standard techniques.

For $1/2<\alpha<1$, the proof is substantially different from the standard approaches: one still obtains uniformly bounded estimates for $\rho_e(L_\beta)$ on vertical lines $\Re(\beta)=\sigma$ even when $\sigma<1/2$, where the function $\beta\mapsto L_\beta$ is unbounded, and this is because the unboundedness in $\beta$ is due to the contribution rank one operator $f\mapsto f(0)\zeta_H(2\beta,z+1)$, which does not contribute to the essential spectral radius. The result is not obtained using a Doeblin-Fortet-Lasota-Yorke inequality.

Question:
Let lpha>0, write lpha=k+b7 with k\in\mathbb N and \eta\in[0,1), and let C^\alpha([0,1]) be the usual Hf6lder space. For \beta\in\mathbb C, define the analytically continued Mayer transfer operator L_\beta on C^\alpha([0,1]) by
\[
L_{\beta}(f)(z)=\sum_{n=0}^k\frac{f^{(n)}(0)}{n!}\,\zeta_H(n+2\beta,z+1)+\sum_{m=1}^\infty \frac{1}{(m+z)^{2\beta}}\,f^*\!\left(\frac{1}{m+z}\right),
\]
where
\[
f^*(z)=f(z)-\sum_{n=0}^k\frac{f^{(n)}(0)}{n!}z^n,
\]
and \zeta_H(s,z)=\sum_{m=0}^\infty (m+z)^{-s} is the Hurwitz zeta function. Let \mathcal L_\gamma denote the corresponding holomorphic Mayer transfer operator
\[
\mathcal L_\gamma(g)(z)=\sum_{m=1}^\infty \frac{1}{(m+z)^{2\gamma}}\,g\!\left(\frac{1}{m+z}\right).
\]
If \sigma_e(L_\beta) denotes the essential spectrum of L_\beta and
\[
\rho_e(L_\beta):=\sup_{\lambda\in\sigma_e(L_\beta)}|\lambda|,
\qquad
\rho(\mathcal L_{\Re(\beta)+\alpha}):=\sup_{\mu\in\sigma(\mathcal L_{\beta+\alpha})}|\mu|,
\]
which statement holds for all \beta with \Re(\beta)>(1-\alpha)/2?

For every \beta with \Re(\beta)>(1-\alpha)/2, the essential spectral radius satisfies
\[
\rho_e(L_\beta)\le \rho(\mathcal L_{\Re(\beta)+\alpha}).
\]
Moreover, for any eigenvalue \lambda of L_\beta with |\lambda|>\rho_e(L_\beta), the corresponding generalized eigenspace consists of analytic functions, and those functions extend to generalized eigenfunctions of \mathcal L_\beta.

Choices:
B.
For every \beta with \Re(\beta)\ge (1-\alpha)/2, the essential spectral radius satisfies
\[
\rho_e(L_\beta)\le \rho(\mathcal L_{\beta+\alpha}).
\]
Moreover, for any eigenvalue \lambda of L_\beta with |\lambda|\ge \rho_e(L_\beta), the corresponding generalized eigenspace consists of analytic functions, and those functions extend to generalized eigenfunctions of \mathcal L_\beta.
C.
For every \beta with \Re(\beta)>(1-\alpha)/2, the essential spectral radius satisfies
\[
\rho_e(L_\beta)\le \rho(\mathcal L_{\Re(\beta)+\alpha}).
\]
D.
For every \beta with \Re(\beta)>(1-\alpha)/2, the essential spectral radius satisfies
\[
\rho_e(L_\beta)< \rho(\mathcal L_{\Re(\beta)+\alpha}).
\]
Moreover, for any eigenvalue \lambda of L_\beta with |\lambda|>\rho_e(L_\beta), the corresponding generalized eigenspace consists of holomorphic functions on a neighborhood of [0,1], and every such generalized eigenfunction extends uniquely to an eigenfunction of \mathcal L_\beta with the same eigenvalue.
E.
For every \beta with \Re(\beta)>(1-\alpha)/2, the essential spectral radius satisfies
\[
\rho_e(L_\beta)\le \rho(\mathcal L_{\Re(\beta)}).
\]
Moreover, for any eigenvalue \lambda of L_\beta with |\lambda|>\rho_e(L_\beta), the corresponding generalized eigenspace consists of analytic functions, and those functions extend to generalized eigenfunctions of \mathcal L_{\Re(\beta)}.

strict half-plane condition and dependence on \Re(\beta) rather than full \beta

analyticity and extension of generalized eigenspaces above the essential spectral radius

generalized eigenfunctions vs eigenfunctions; non-strict radius inequality strengthened to strict and extension target sharpened

loss of the +\alpha shift coming from Hölder regularity in the comparison operator


\label{thm:gawron-generalization}
	If $z\in \mathcal S$, then $|z| > \tfrac 14$.

\begin{thm}\label{thm:MO-ineq}
	For all real numbers $t$ and all positive integers $n$,
	\begin{equation}\label{eqn:tao-ineq}
		\abs{\left(1+\frac{it}n\right)^n - 1} \geq \big|e^{it} - 1\big| = 2\sin\frac t2.
	\end{equation}
\end{thm}

\begin{thm}\label{thm:roots-disk}
	All roots of $\stern n\lm$ lie outside the disk $\{\abs{z-2}\leq 1\}\subseteq\C$.
\end{thm}

\begin{thm}\label{thm:gawron-generalization}
	If $z\in \calS$, then $|z| > \tfrac 14$.
\end{thm}

\begin{conj}\label{conj:location-roots}
	All elements of $\calS$ lie in the half-plane $\{\Re w < 1\}$.
\end{conj}

\begin{figure}[ht]
	\centering
	\captionsetup{width=0.75\linewidth}
	\includegraphics[scale=0.75]{sternpoly-roots-deg-21-BW}
	\caption{\small The zeros of $\stern n\lm$ in the range $\{a+bi:-4\leq a\leq 1, |b|\leq 3\}$, where $1\leq n<2^{21}$ is odd.}
	\label{fig:stern-roots}
\end{figure}

\begin{corr}\label{thm:prime-poly-irred}
	For each prime number $p$, the Stern polynomial $\stern p \lm$ is irreducible in $\Q$.
\end{corr}

The Stern sequence $(s_n)_{n\geq 0}$ of positive integers, named after Moritz Abraham Stern, is given by $s_0 = 0$, $s_1 = 1$, and
\begin{equation}\label{eqn:stern-seq}
 s_{2n} = s_n,\quad s_{2n+1} = s_n + s_{n+1}.
\end{equation}
There is a vast body of literature (e.g. \cite{Coons2014,Northshield2010,Reznick2008,Stanley2019} and the references therein) about the Stern sequence. Perhaps the most striking property of the Stern sequence is that every positive rational number appears exactly once in the sequence $(s_{n+1}/s_n)_{n\geq 1}$, giving an explicit enumeration of $\mathbb Q^+$. Stern proved this fact 15 years before Cantor introduced the notion of a countable set!

\par In \cite{Klavzar2007}, Klav\v{z}ar, Milutinovi\'{c}, and Petr define a polynomial analogue $\stern n\lambda$ of the Stern sequence via $\stern 0\lambda = 0$, $\stern 1\lambda = 1$, and
\begin{equation}\label{eqn:stern-def}
 \stern{2n}\lambda = \lambda \stern n\lambda,\quad \stern{2n+1}\lambda = \stern n\lambda + \stern{n+1}\lambda.
\end{equation}
The first $16$ terms of the sequence $\stern n\lambda$ are given in Table \ref{table:stern-16}. This table suggests many patterns, several of which are recorded in Proposition \ref{prop:basic-patterns}.

\begin{table}[ht]
 \input{table-stern-16}
 \caption{\small The first $16$ terms of the sequence $\stern n\lambda$.}
 \label{table:stern-16}
\end{table}

While there are several papers which analyze the sequence $\stern n\lambda$ (see \cite{Dilcher2017,Dilcher2018,Gawron2014,Ulas2011} and the references therein), not much is known about the zeros of Stern polynomials. Because $\stern{2n}\lambda = \lambda \stern n \lambda$ for all $n\geq 1$, it suffices to examine the roots of $\stern n\lambda$ only when $n$ is odd. Define
\[
\mathcal S\coloneqq \{z\in\mathbb C: \stern n z = 0\text{ for some odd }n\geq 1\}.
\]
In \cite[Theorem 2.1]{Gawron2014}, Gawron proves that $0$, ${-}1$, ${-}\tfrac12$, and $-\tfrac13$ are the only rational zeros of any Stern polynomial by showing, more generally, that any real number in the closed interval $[-\tfrac14,\tfrac14]$ cannot be in $\mathcal S$. His methods extend easily to show that any complex number $z$ with $|z| \leq \tfrac14$ cannot be in $\mathcal S$. For completeness, we record the proof here.

\begin{prop}\label{prop:basic-patterns}
	Let $n$ be a nonnegative integer.

	\vspace{-3pt}

	\begin{enumerate}
		\item $\stern{n}{0} = 0$ when $n$ is even, and $\stern n 0 = 1$ when $n$ is odd.
		\item $\stern{n}{1} = s_n$, where $s_n$ is the Stern sequence \eqref{eqn:stern-seq}.
		\item $\stern{n}{2} = n$.
		\item When $n = 2^r$ for a nonnegative integer $r$, $\stern n \lm = \lm^r$.
		\item When $n = 2^r - 1$ for a nonnegative integer $r$, $\stern n\lm = (\lm)_r$.
	\end{enumerate}
\end{prop}

\begin{table}[ht]
	\input{table-stern-16}
	\caption{\small The first $16$ terms of the sequence $\stern n\lm$.}
	\label{table:stern-16}
\end{table}

While there are several papers which analyze the sequence $\stern n\lm$ (see \cite{Dilcher2017,Dilcher2018,Gawron2014,Ulas2011} and the references therein), not much is known about the zeros of Stern polynomials. Because $\stern{2n}\lm = \lm \stern n \lm$ for all $n\geq 1$, it suffices to examine the roots of $\stern n\lm$ only when $n$ is odd. Define
\[
\calS\coloneqq \{z\in\C: \stern n z = 0\text{ for some odd }n\geq 1\}.
\]
In \cite[Theorem 2.1]{Gawron2014}, Gawron proves that $0$, ${-}1$, ${-}\tfrac12$, and $-\tfrac13$ are the only rational zeros of any Stern polynomial by showing, more generally, that any real number in the closed interval $[-\tfrac14,\tfrac14]$ cannot be in $\calS$. His methods extend easily to show that any complex number $z$ with $|z| \leq \tfrac14$ cannot be in $\calS$. For completeness, we record the proof here.

\begin{proof}
 Let $z$ be any complex number with $|z| \leq \tfrac 14$. Let $b_n\coloneqq |\stern n z|$ for each positive integer $n$. We show more strongly that
 \begin{equation}\label{eqn:stronger-statement}
 b_{2n+1} > \tfrac12\max\{b_n,b_{n+1}\} > 0
 \end{equation}
 for all $n\geq 1$.

\begin{thm}[Parabola Theorem, {\cite[Theorem 3.43]{Lorentzen2008}}]\label{thm:parabola}
 For fixed $|\alpha| < \tfrac\pi 2$, let
 \[
 V_\alpha\coloneqq -\tfrac12 + e^{i\alpha}\overline{\mathbb H} = \{w\in\C: \Re(we^{- i\alpha})\geq -\tfrac12\cos\alpha\}
 \]
 and
 \[
 E_\alpha \coloneqq \{a\in\C: |a| - \Re(ae^{-2i\alpha})\leq \tfrac12\cos^2\alpha\}.
 \]
 Then $E_\alpha$ is the element set for continued fractions $\boldsymbol{K}(a_n | 1)$ corresponding to the value set $V_\alpha$.
\end{thm}
The region $V_\alpha$ is a half-plane whose boundary is a line intersecting the real axis at $z = -\tfrac12$. In particular, because $|\alpha| < \tfrac\pi 2$, $V_\alpha$ contains the half-line $[-\tfrac12,\infty)$. The boundary of $E_\alpha$ is a parabola with focus at the origin and vertex at $-\tfrac14 e^{2i\alpha}\cos^2\alpha$. This parabola intersects the real axis at $z = -\tfrac14$.

We now tie the continued fraction theory back to Stern polynomials.
Recall that Corollary \ref{thm:stern-cont-frac} expresses the ratio
of two Stern polynomials as a continued fraction with elements in $\C$.
This form on its own is difficult to work with. We instead note (e.g. \cite[Corollary 2.15]{Lorentzen2008}) that any continued fraction $\boldsymbol{K}(a_n|b_n)$ with $b_j\neq 0$ is equivalent to the continued fraction $\boldsymbol{K}(c_n|1)$, where $c_j = \tfrac{a_j}{b_jb_{j+1}}$. As an explicit example,
\begin{equation}\label{eqn:cfrac-example}
 \frac{\stern{[[2,3,5]]}\lm}{\stern{[[3,5]]}\lm} = (\lm)_2 + \cfrac{\lm^2}{(\lm)_3 + \cfrac{\lm^3}{(\lm)_5}}
 = (\lm)_2 \left[1 + \cfrac{\dfrac{\lm^2}{(\lm)_2(\lm)_3}}{1 + \cfrac{\lm^3}{(\lm)_3(\lm)_5}}\right].
\end{equation}
\begin{remark}
 While continued fractions of the form $\boldsymbol{K}(1|d_n)$ are more common, the coefficients $d_n$ are significantly messier than the coefficients $c_n$ (see \cite[Corollary 2.15]{Lorentzen2008}). We opt to use the less-common form $\boldsymbol{K}(c_n|1)$ to simplify our analysis.
\end{remark}
Equation \eqref{eqn:cfrac-example} suggests that the ratios $\tfrac{z^a}{(z)_a(z)_b}$, where $a$ and $b$ are any positive integers, may play some importance.
For this reason, we define $z_{a,b}\coloneqq \tfrac{z^a}{(z)_a(z)_b}$ and
\[
\calA_z\coloneqq \left\{z_{a,b}:(a,b)\in\N^2\right\}.
\]
Suppose $\calA_z\subseteq E_\alpha$ for some angle $\alpha$ with $|\alpha| < \tfrac{\pi}2$.
Then $V_\alpha$ is a value set for the continued fraction \eqref{eqn:stern-cont-frac} at $\lm = z$. In particular, $\infty$ is not a possible value for this fraction, so $\stern n z \neq 0$ for any $z$.

\begin{itemize}
 \item First suppose $x\in[0,\pi]$. Because both $\sin x$ and $x - \tfrac{x^2}{\pi}$ are symmetric about the axis $x = \tfrac\pi 2$, it suffices to prove the inequality for $x\in[0,\tfrac\pi 2]$. Within this smaller interval, cosine is concave down with $\cos 0 = 1$ and $\cos\tfrac \pi 2 = 0$, so
 \begin{equation}\label{eqn:cos-ineq}
 \cos x \geq 1 - \frac{2x}{\pi}.
 \end{equation}
 Integrating both sides of \eqref{eqn:cos-ineq} yields \eqref{ineq-sinc} in this case.
 \item Now suppose $x\notin[0,\pi]$. By symmetry, it suffices to prove the inequality for $x\geq \pi$. In this interval, $1 - \tfrac{2x}{\pi} \leq -1 \leq \cos x$. It follows that \eqref{eqn:cos-ineq}, and thus \eqref{ineq-sinc}, holds as well.
 \end{itemize}
\end{proof}

\begin{thm}\label{thm:min-geom-series}
 Let $n$ be a positive integer, and suppose $z\in\calB$. Then $|(z)_n| \geq \min(n,\tfrac{11}2)$.
\end{thm}

Now let
 \[
 M\coloneqq 2\pi\left(1 - \frac{5.5}{n}\right) = \pi\left(2 - \frac{11}n\right).
 \]
 There are two cases to consider. First, suppose $t\leq M$. Then
 \[
 n\left(1 - \frac{t}{2\pi}\right) \geq n\left(1 - \frac{M}{2\pi}\right) = \frac{11}2.
 \]
 Now suppose $t\geq M$. Because $n\geq 15$, the coefficient $\tfrac{1}{2n} - \tfrac{11}{8n^2}$ is positive, so
 \begin{equation}\label{eqn:asymp-w-tn}
 t + t^3\left(\frac{1}{2n} - \frac{11}{8n^2}\right) \geq \pi\left(2 - \frac{11}n\right) + \pi^3\left(2 - \frac{11}n\right)^3\left(\frac{1}{2n} - \frac{11}{8n^2}\right).
 \end{equation}
 To estimate \eqref{eqn:asymp-w-tn}, let $x = \tfrac 1n$, so it suffices to analyze the polynomial
 \[
 P(x) := \pi(2-11x) + \pi^3(2-11x)^3(\tfrac 12x - \tfrac{11}{8}x^2).
 \]
 The polynomial $P$ has exactly one critical point in the interval $[0,\tfrac 1{15}]$, occurring at $x = x_0 \approx 0.024$. In particular, $P$ is increasing on $[0,x_0]$ and decreasing on $[x_0,\tfrac1{15}]$. Compute
 \[
 P(0) = 2\pi\quad\text{and}\quad P\left(\frac 1{15}\right) = \frac{19 \pi }{15}+\frac{336091 \pi ^3}{6075000} \approx 5.695 > \frac{11}2.
 \]
 It follows that $P(x) > \tfrac{11}2$ for all $x\in[0,\tfrac{1}{15}]$, and thus $|w_{n,t}^n - 1| \geq M|w_{n,t}-1|$ for all $t\in[0,2\pi]$ and $n\geq 15$.
\end{proof}

\begin{prop}\label{prop:silli}
 Let $z\in\calB^+$. Then $0\leq \arg z \leq \tfrac\pi 6$ and
 \begin{equation}\label{eqn:re-ineq}
 |\Re z^{-4}|\leq |\Re z^{-2}|.
 \end{equation}
\end{prop}

\begin{corr}[{\cite[Theorem 1]{Schinzel2014}}]\label{thm:stern-cont-frac}
	Let $a_1,\ldots, a_t$ be positive integers, where $t\geq 2$.  Then
	\begin{equation}\label{eqn:stern-cont-frac}
		\frac{\stern{[[a_1,\ldots, a_t]]}{\lm}}{\stern{[[a_2,\ldots, a_t]]}{\lm}}
		=
		(\lm)_{a_1} + \cfrac{\lm^{a_1}}{(\lm)_{a_2} + \cfrac{\lm^{a_2}}{\cdots + \cfrac{\cdots}{(\lm)_{a_{t-1}} + \cfrac{\lm^{a_{t-1}}}{(\lm)_{a_t}}}}}\,.
	\end{equation}
\end{corr}

\begin{proof}
	Let $z$ be any complex number with $|z| \leq \tfrac 14$.  Let $b_n\coloneqq |\stern n z|$ for each positive integer $n$.  We show more strongly that
	\begin{equation}\label{eqn:stronger-statement}
		b_{2n+1} > \tfrac12\max\{b_n,b_{n+1}\} > 0
	\end{equation}
	for all $n\geq 1$.

	\par The proof of \eqref{eqn:stronger-statement} proceeds by induction on $n$.  The base case, $n = 1$, follows because
	\[
	b_3 = |z + 1| \geq \tfrac 34 > \tfrac 12
	= \tfrac 12\max\{1, |z|\} = \tfrac 12\max\{b_1,b_2\}.
	\]

	\par There are two cases to consider.  First assume $n=2k$ is even.  Then
	\begin{align*}
		b_{4k+1} &= |\stern{4k+1}z| = |\stern{2k+1} z + \stern{2k} z|\\
		&= |z \stern kz + \stern{2k+1} z| \geq b_{2k+1} - \tfrac14 b_k\\
		&\geq b_{2k+1} - \tfrac12 b_{2k+1} = \tfrac12 b_{2k+1}.
	\end{align*}
	Moreover, $b_{2k+1} \geq \frac12 b_k$, and thus $\max\{b_{2k},b_{2k+1}\} = b_{2k+1} > 0$.  In this case, our inequality is proved.

	\par Now assume $n = 2k + 1$ is odd.  Then
	\begin{align*}
		b_{4k+3} &= |\stern{4k+3}z| = |\stern{2k+1} z + \stern{2k+2} z|\\
		&= |\stern{2k+1} z + z\stern{k+1} z| \geq b_{2k+1} - \frac 14 b_{k+1}\\
		&\geq b_{2k+1} - \tfrac12 b_{2k+1} = \tfrac12 b_{2k+1}.
	\end{align*}
	Moreover, $b_{2k+1} \geq \frac12 b_{k+1}$, and so in this case we also have $\max\{b_{2k},b_{2k+1}\} = b_{2k+1} > 0$.  We have exhausted both cases, completing the proof of Theorem \ref{thm:gawron-generalization}.
\end{proof}

In \cite{Dilcher2017}, Dilcher et. al. focus more specifically on the complex roots of $\stern n\lambda$.  Their paper makes the following conjecture.

\begin{conj}\label{conj:location-roots}
	All elements of $\mathcal S$ lie in the half-plane $\{\operatorname{Re} w < 1\}$.
\end{conj}

By generalizing the Enestrom-Kakeya theorem, they prove Conjecture \ref{conj:location-roots} for several classes of positive integers $n$ taking the form $2^n\pm k$, where $k$ is fixed and $2^n \geq k$.  These are the only two papers the author could find which discuss the complex zeros of $\stern n\lambda$.

\par Figure \ref{fig:stern-roots} shows a snapshot of $\mathcal S$.  One striking feature of this figure is the contrasting behavior of these roots within the half-planes $\{\operatorname{Re} w\geq 0\}$ and $\{\operatorname{Re} w < 0\}$.  These differences present difficulties in fully characterizing the geometry of $\mathcal S$.

\begin{figure}[ht]
	\centering
	\captionsetup{width=0.75\linewidth}
	\includegraphics[scale=0.75]{sternpoly-roots-deg-21-BW}
	\caption{\small The zeros of $\stern n\lambda$ in the range $\{a+bi:-4\leq a\leq 1, |b|\leq 3\}$, where $1\leq n<2^{21}$ is odd.}
	\label{fig:stern-roots}
\end{figure}

In this paper, we partially resolve Conjecture \ref{conj:location-roots} by establishing the following result.

\begin{thm}\label{thm:roots-disk}
	All roots of $\stern n\lambda$ lie outside the disk $\{\left|z-2\right|\leq 1\}\subseteq\mathbb C$.
\end{thm}

To prove Theorem \ref{thm:roots-disk}, we use a continued fraction representation for a ratio of Stern polynomials (Theorem \ref{thm:stern-cont-frac}) independently discovered by Reznick \cite{Reznick2008} and Schinzel \cite{Schinzel2014}.  This allows us to use the Parabola Theorem (Theorem \ref{thm:parabola}) to show that, for certain values of $z\in\mathbb C$, the denominators of these continued fractions can never be zero.  Along the way, we establish inequalities in $\mathbb C$ relating to the sums $1 + z + \cdots + z^{n-1}$ which may be of independent interest.  The most notable of these inequalities is Theorems \ref{thm:MO-ineq}, which proves a lower bound for this geometric series whenever $\operatorname{Re} z \geq 1$.

As a corollary, we obtain the following surprising fact, resolving a conjecture of Ulas and Ulas (\cite{Ulas2011}).

\begin{corr}\label{thm:prime-poly-irred}
	For each prime number $p$, the Stern polynomial $\stern p \lambda$ is irreducible in $\mathbb Q$.
\end{corr}
Ulas and Ulas had verified this conjecture computationally for the first million primes $p$.  Additionally, Schinzel in \cite{Schinzel2011} proved Corollary \ref{thm:prime-poly-irred} for all primes $p < 2017$ by using finite differences to bound the leading coefficient of any proper divisor of $\stern n\lambda$.  However, these previous attempts to prove the conjecture were algebraic in nature, whereas our proof depends on the analytic properties of $\mathcal S$.

\paragraph{Acknowledgments.} This paper is adapted from the author's PhD dissertation \cite{Altizio2025}.  The author thanks his advisor, Dr. Bruce Reznick, for helpful correspondence.

Let $z$ be any complex number with $|z|\leq \tfrac14$ and set $b_n\coloneqq |\stern n z|$. The proof shows the stronger claim
\begin{equation}\label{eqn:stronger-statement}
 b_{2n+1} > \tfrac12\max\{b_n,b_{n+1}\} > 0
\end{equation}
for all $n\ge1$, and proceeds by induction on $n$.

Base case $n=1$:
\[
 b_3=|z+1|\ge \tfrac34>\tfrac12=\tfrac12\max\{1,|z|\}=\tfrac12\max\{b_1,b_2\}.
\]

Inductive step splits into two cases.

(i) If $n=2k$ is even, then
\[
 b_{4k+1}=|\stern{4k+1}z|=|\stern{2k+1}z+\stern{2k}z|=|z\,\stern k z+\stern{2k+1}z|
 \ge b_{2k+1}-\tfrac14 b_k\ge b_{2k+1}-\tfrac12 b_{2k+1}=\tfrac12 b_{2k+1}.
\]
Moreover, $b_{2k+1}\ge \tfrac12 b_k$, hence $\max\{b_{2k},b_{2k+1}\}=b_{2k+1}>0$, giving \eqref{eqn:stronger-statement}.

(ii) If $n=2k+1$ is odd, then
\[
 b_{4k+3}=|\stern{4k+3}z|=|\stern{2k+1}z+\stern{2k+2}z|=|\stern{2k+1}z+z\,\stern{k+1}z|
 \ge b_{2k+1}-\tfrac14 b_{k+1}\ge b_{2k+1}-\tfrac12 b_{2k+1}=\tfrac12 b_{2k+1}.
\]
Moreover, $b_{2k+1}\ge \tfrac12 b_{k+1}$, so again $\max\{b_{2k},b_{2k+1}\}=b_{2k+1}>0$. These two cases exhaust the induction, completing the proof of Theorem~\ref{thm:gawron-generalization}.

Question:
For the Stern polynomials \(\operatorname{St}_n(\lambda)\) defined by \(\operatorname{St}_0(\lambda)=0\), \(\operatorname{St}_1(\lambda)=1\), and for all integers \(n\ge 0\),
\[
\operatorname{St}_{2n}(\lambda)=\lambda\,\operatorname{St}_n(\lambda),\qquad
\operatorname{St}_{2n+1}(\lambda)=\operatorname{St}_n(\lambda)+\operatorname{St}_{n+1}(\lambda),
\]
define
\[
\mathcal S:=\{z\in\mathbb C: \operatorname{St}_n(z)=0\text{ for some odd }n\ge 1\}.
\]
Which quantitative estimate holds for every \(z\in\mathcal S\)?

Every \(z\in\mathcal S\) satisfies \(|z|>\tfrac14\).

Choices:
B. Every \(z\in\mathcal S\) satisfies \(|z|\ge \tfrac14\).
C. Every \(z\in\mathcal S\) satisfies \(|z|>0\).
D. Every \(z\in\mathcal S\) satisfies \(\Re(z)>-\tfrac14\).
E. Every \(z\in\mathcal S\) satisfies \(|z|>\tfrac13\).


\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 \(N(u)=0\).