2.1 Graph approximations

We begin with a discussion of graph approximations following Strichartz [Reference Strichartz15].

A self-similar set K is defined to be the unique nonempty compact set satisfying:

(2.1) $$ \begin{align} K=\bigcup_{j=1}^N \varphi_j(K), N\geq 2 \end{align} $$

for some distinct injective contraction mappings $\varphi _j$ on a complete metric space. Equation (2.1) is known as a self-similar identity. Here, K is self-similar in the sense that it is a union of copies of itself.

To illustrate, we take the unit interval I as an example. The mappings

(2.2) $$ \begin{align} F_0(x)=\frac{x}{2}, F_1(x)=\frac{x}{2} + \frac{1}{2} \end{align} $$

on I, with fixed points $0$ and $1,$ respectively, give the self-similar identity

(2.3) $$ \begin{align} I=F_0I \cup F_1I. \end{align} $$

More generally, by composing the maps $F_i$ , we obtain

(2.4) $$ \begin{align} I=\bigcup_{|w|=m}F_wI, w=(w_1,\dots, w_m), w_j\in \{0,1\}, \end{align} $$

where $F_w=F_{w_1}\circ \dots \circ F_{w_m}$ . We call w a word of length $m=|w|$ , $F_wI$ an m-level cell, and write $x\underset {m}\sim y$ if there is an m-level cell with x and y as its boundary points. Let $V_0=\{q_0,q_1\}, q_0=0,q_1=1$ , and define inductively

(2.5) $$ \begin{align} V_m := \bigcup_{i\in\{0,1\}}F_iV_{m-1}. \end{align} $$

We regard the sets $V_m$ as the vertices of the graph $\Gamma _m$ , with edge relation $x\underset {m}\sim y$ , built inductively from $\Gamma _{m-1}$ by taking the two images $F_0\Gamma _{m-1}$ and $F_1\Gamma _{m-1}$ . Observe that $V_m$ increases to $V_*$ , the set of dyadic rationals in I, which is dense in I. Hence, if we are interested only in continuous functions on I, then it suffices to know their values on  $V_*$ .

The case of $SG$ may be viewed as a generalization of the case of the unit interval (I) with

(2.6) $$ \begin{align} F_i(x)=\frac{1}{2}(x-q_i) +q_i, i=0,1,2, \end{align} $$

where the points $q_i$ are the vertices of a nondegenerate triangle in the plane. $SG$ satisfies the self-similar identity:

(2.7) $$ \begin{align} SG=\bigcup_{i=0}^2F_i(SG). \end{align} $$

In the case of $SG$ , we take $V_0 = \{q_0,q_1,q_2\}$ , the vertices of the triangle, and construct a graph $\Gamma _m$ with vertices $V_m$ by defining the edge relation $x\underset {m}\sim y$ if there is a cell of level m containing both x and y. $\Gamma _m$ is obtained by taking three copies $F_i\Gamma _{m-1}$ of $\Gamma _{m-1}$ and identifying the points $\{F_iq_j = F_jq_i\}_{i\neq j}, i,j=0,1,2$ .

2.2 Spectral decimation

The precise definition of spectral decimation can be found in [Reference Derfel, Grabner and Vogl4]. For the purpose of this article, it suffices to describe it as follows.

Let $V_m$ denote the vertices of the m-level approximation of K. We define the discrete Laplacian $\Delta _m$ on $V_m$ by

(2.8) $$ \begin{align} \Delta_mu(x) := \sum_{y\in V_m,x\underset{m}\sim y}(u(y)-u(x)), x \in V_m. \end{align} $$

It is known, for example, from [Reference Hare, Steinhurst, Teplyaev and Zhou7] that for a fully symmetric self-similar structure on a finitely ramified fractal K, there is a function R, called the spectral decimation function relating the eigenvalues of $\Delta _m$ and those of $\Delta _{m-1}$ . In [Reference Shima13, Proposition 2.18], Shima shows that $R(0)=0$ and $c_\Delta := R'(0)>1$ . Thus, $0$ is a repulsive fixed point of the spectral decimation function.

Definition 2.1 Define the standard Laplacian on K as

(2.9) $$ \begin{align} \Delta u(x) = \lim_{n\to\infty} c_\Delta^n \Delta_nu(x), u \in C(K) \end{align} $$

if the limit exists.

Let $\phi _0,\dots , \phi _L$ be the partial inverses of R with $\phi _0$ satisfying $\phi _0(0)=0$ .

If all the eigenvalues of $\Delta $ have the form

(2.10) $$ \begin{align} \lambda_k = \lim_{n\to\infty} c_{\Delta}^n\lambda_k^{(n)} = \lim_{n\to\infty} c_{\Delta}^{n_0 +n}\phi_0^n\lambda_k^{(n_0)}, k \in \mathbb{N,} \end{align} $$

where $\lambda _k^{(n)}$ is the k-th eigenvalue of $\Delta _n$ , then $\Delta $ is said to admit spectral decimation [Reference Hare, Steinhurst, Teplyaev and Zhou7].

The class of fully symmetric self-similar finitely ramified fractals admits spectral decimation.

Note that Definition 2.1 describes a phenomenon where a suitable sequence of eigenvalues in the finite-level approximation of a self-similar set is produced by an orbit of a particular dynamical system. This phenomenon is what is termed spectral decimation [Reference Fukushima and Shima6].

3.1 Zero infimum spacing

The next theorem gives a condition under which the infimum spacing is zero. Here, our main result is Theorem 3.1, where we show that the existence of a nonzero repulsive fixed point of the spectral decimation function with a multiplier larger than the zero fixed point leads to a zero infimum spacing in the spectrum.

Proof Let $\phi _\zeta $ be the inverse branch of R with $\zeta $ in its range. Note that given ${\delta>0}$ , there is a neighborhood $U_\delta $ of $\zeta $ , such that $\phi _\zeta ^m$ is a contraction on $U_\delta $ with the contraction rate bounded by $(\vert {R'(\zeta )}\vert -\delta )^{-m}$ . Moreover, since $\zeta $ is in the Julia set of R and is an attracting fixed point of $\phi _\zeta $ , we may choose n such that $x_1, x_2 \in \sigma (\Delta _n)\cap U_\delta $ . Now, by [Reference Shima13, Proposition 3.1],

$$\begin{align*}c_\Delta^{n+j+m}\phi_0^j(\phi_\zeta^m(x_1)), \ \ c_\Delta^{n+j+m}\phi_0^j(\phi_\zeta^m(x_2))\end{align*}$$

converge to eigenvalues of $\Delta $ say $\lambda ^{(m)},(\lambda ^{(m)})'$ , respectively, as $j\to \infty $ . And by the Kœnigs linearization theorem (see, e.g., [Reference Milnor10, Theorem 6.1]), there is a local holomorphic change of coordinates $\mathscr {R}$ , called the Kœnigs function, such that

$$\begin{align*}\lambda^{(m)}=c^{n+m}\mathscr{R}(\phi_\zeta^m(x_1)), \ \ (\lambda^{(m)})'=c_\Delta^{n+m}\mathscr{R}(\phi_\zeta^m(x_2)). \end{align*}$$

Hence, for some $\gamma _m$ lying between $\phi _\zeta ^m(x_1)$ and $\phi _\zeta ^m(x_2),$ the spacing between these two eigenvalues is bounded by

$$ \begin{align*} \vert{\lambda^{(m)}-(\lambda^{(m)})'}\vert&=c_\Delta^n\vert{\mathscr{R}'(\gamma_m)}\vert c_\Delta^m\vert{(\phi_\zeta^m(x_1)-\phi_\zeta^m(x_2))}\vert\\ &\leq c_\Delta^n\vert{\mathscr{R}'(\gamma_m)}\vert\left(\frac{c_\Delta}{\vert{R'(\zeta)}\vert-\delta}\right)^m\vert{x_1-x_2}\vert, \end{align*} $$

which gets arbitrarily small as $m\to \infty $ for small $\delta $ .

Figure 1

$SG_3$ .

Figure 1 Long description

The diagram depicts a Sierpinski triangle, a fractal based on an equilateral triangle.

* The overall structure is a large equilateral triangle with its apex pointing upward.

* At the first level of recursion, the central area is a large white inverted equilateral triangle, which divides the original black triangle into three smaller black equilateral triangles located at the top, bottom-left, and bottom-right corners.

* At the second level, each of those three black triangles contains its own smaller white inverted triangle at its center, further subdividing them into nine even smaller black triangles.

* This recursive process continues through multiple iterations. Each black triangular area is composed of three smaller triangles around a central white void.

* At the smallest visible scale, the black areas appear as tiny dots or triangles, creating a complex, self-similar lattice pattern where the large-scale geometry is mirrored exactly in the microscopic details.

In [Reference Bajorin, Chen, Dagan, Emmons, Hussein, Khalil, Mody, Steinhurst and Teplyaev3], Bajorin et al. have described the spectrum of the discrete Laplacian on $SG_3$ (Figure 1), which is given by

$$\begin{align*}\sigma(\Delta_n) = (R^{-n}(0))\bigcup\Bigg(R^{-(n-1)}\Bigg(\frac{3 \pm \sqrt{5}}{4}\Bigg)\Bigg)\bigcup \Bigg\{\frac{3}{2}\Bigg\}, n\geq 0\end{align*}$$

and whose spectral decimation function is

$$\begin{align*}R(x)=\frac{6x(x-1)(4x-5)(4x-3)}{6x-7}\end{align*}$$

with derivative,

$$\begin{align*}R'(x) = \frac{6(288x^4-1024x^3+1290x^2-658x+105)}{(6x-7)^2}.\end{align*}$$

Note that $0,1,\frac {3}{4},\frac {5}{4} \in R^{-n}(0) \subseteq \sigma (\Delta _n)$ . R also has a fixed point between $1.30$ and $1.31$ . A direct computation shows $R'(0)=\frac {90}{7} < 23.542849 \approx \inf _{y \in [1.2,1.5]}R'(y)$ . Hence, by Theorem 3.1, we must have that the infimum spacing in $\sigma (\Delta )$ is zero.

Theorem 3.3 Let $\{\lambda _{n,k}\}_k\subseteq \sigma (\Delta _n)$ be the complete list of distinct finite-level eigenvalues of $\Delta _n$ in increasing order. Then,

$$\begin{align*}\inf\{\vert{\lambda-\lambda'}\vert:\lambda\neq\lambda',\lambda,\lambda'\in\sigma(\Delta)\}=0 \end{align*}$$

if and only if

$$\begin{align*}\lim_n\min_k\frac{\vert{\lambda_{n,k}-\lambda_{n,k+1}}\vert}{\vert{\lambda_{n,0}-\lambda_{n,1}}\vert}=0. \end{align*}$$

Proof Using the notation in the previous theorem, note that for each n and k, we have that $c_\Delta ^n\mathscr {R}(\lambda _{n,k})$ and $c_\Delta ^n\mathscr {R}(\lambda _{n,k+1})$ are eigenvalues of $\Delta $ . Moreover,

$$ \begin{align*} \min_k\vert{c_\Delta^n\mathscr{R}(\lambda_{n,k})- c_\Delta^n\mathscr{R}(\lambda_{n,k+1})}\vert&=\min_k\vert{\mathscr{R}'(\gamma_{n,k})}\vert c_\Delta^n\vert{\lambda_{n,k}-\lambda_{n,k+1}}\vert\\ &=c_\Delta^n\vert{\lambda_{n,0}-\lambda_{n,1}}\vert\min_k\vert{\mathscr{R}'(\gamma_{n,k})}\vert\frac{\vert{\lambda_{n,k}-\lambda_{n,k+1}}\vert}{\vert{\lambda_{n,0}-\lambda_{n,1}}\vert}. \end{align*} $$

Since $\mathscr {R}'$ is bounded and the factor $c_\Delta ^n\vert {\lambda _{n,0}-\lambda _{n,1}}\vert $ converges to $\mathscr {R}(x)-\mathscr {R}(y)$ for some $x,y$ , the reverse implication follows by taking the limit as $n\to \infty $ .

On the other hand, if $\inf _{\lambda \neq \lambda ', \lambda ,\lambda '\in \sigma (\Delta )}\vert {\lambda -\lambda '}\vert =0$ , then for each $m\in \mathbb {N}$ , there exist $\lambda ,\lambda '\in \sigma (\Delta )$ such that $\vert {\lambda -\lambda '}\vert \leq \frac {1}{m}$ . Now note that for any two distinct eigenvalues, $\lambda ,\lambda '\in \sigma (\Delta )$ , we may choose $n\geq m$ sufficiently large such that $\lambda =c_\Delta ^n\mathscr {R}(\lambda _{n,j}), \lambda '=c_\Delta ^n\mathscr {R}(\lambda _{n,j'})$ , and we have

$$ \begin{align*} \frac{1}{m}&\geq \vert{\lambda-\lambda'}\vert\\ &=\vert{c_\Delta^n\mathscr{R}(\lambda_{n,j})-c_\Delta^n\mathscr{R}(\lambda_{n,j'})}\vert\\ &=\vert{\mathscr{R}'(\gamma_{n,j})}\vert c_\Delta^n\vert{\lambda_{n,j}-\lambda_{n,j'}}\vert\\ &\geq \vert{\mathscr{R}'(\gamma_{n,j})}\vert c_\Delta^n\vert{\lambda_{n,0}-\lambda_{n,1}}\vert\min_k\frac{\vert{\lambda_{n,k}-\lambda_{n,k+1}}\vert}{\vert{\lambda_{n,0}-\lambda_{n,1}}\vert}\\ &\geq C\min_k\frac{\vert{\lambda_{n,k}-\lambda_{n,k+1}}\vert}{\vert{\lambda_{n,0}-\lambda_{n,1}}\vert}. \end{align*} $$

The conclusion follows by letting n go to infinity.

3.2 Positive infimum spacing

The proof of the next result relies on the mean value theorem and induction. It is crucial for proving the main Theorem 3.5, which guarantees that the infimum spacing in the spectrum is positive under certain conditions. While it has many technical assumptions, to the author’s knowledge, all of them – except that $R'(0)$ is maximal – are satisfied in all examples of spectral decimation considered in [Reference Bajorin, Chen, Dagan, Emmons, Hussein, Khalil, Mody, Steinhurst and Teplyaev3, Reference Shima13].

Lemma 3.4 Assume $\Delta $ admits spectral decimation with spectral decimation function R, and suppose $0$ is a fixed point of R with $\vert {R'(0)}\vert>1$ .

Let $D_0=\{0,x_1,\dots ,x_r\}$ be a finite set with $0<x_1<\cdots <x_r$ . Suppose the interval $[0,x_r]$ is a backward invariant under R and the inverse branches $\phi _j$ of R are defined on all of this interval.

Assume further that

• • $R^{-1}(0,x_r)\cap D_0 =\emptyset , \ \ R^{-1}\{0,x_r\}\subseteq D_0$ and

• • $R'(0) = \max _{x\in R^{-1}[0,x_r]}|R'(x)|$ ,

and set $D_n:= \cup _{m= 0}^n R^{-m}(D_0)$ . Then, there is a constant $C_0$ , depending on $D_0$ , such that

$$\begin{align*}min_{x,y\in D_n, x\neq y}|x-y|\geq \frac{C_0}{(R'(0))^n}. \end{align*}$$

Proof Let $C_0 = min_k |x_k -x_{k-1}|, x_j \in D_0$ . We know $D_1= D_0 \cup R^{-1}(D_0)$ . If $x,y \in D_0, x\neq y$ , then we obtain

$$\begin{align*}|x-y| \geq C_0 \geq \frac{C_0}{R'(0)}. \end{align*}$$

If $x \in D_0$ and $y\in R^{-1}(D_0)$ , then $y\in R^{-1}[0,x_r]$ . Thus, $y\in \phi _k[0,x_r]$ , for some k. But $R^{-1}(0,x_r)\cap D_0 = \emptyset , R^{-1}\{0,x_r\} \subseteq D_0$ implies $\vert {x-y}\vert \geq \min _{z\in \{0,x_r\}}\vert {y-\phi _k(z)}\vert $ . For $z\in \{0,x_r\}$ , we have $\left |\frac {R(y)-R(\phi _k(z))}{y-\phi _k(z)}\right |=\vert {R'(c)}\vert \leq R'(0)$ , for some $c\in \phi _k[0,x_r]$ . Thus, $\vert {y-\phi _k(z)}\vert \geq \frac {\vert {R(y)-R(\phi _k(z))}\vert }{R'(0)}\geq \frac {C_0}{R'(0)}.$

For $x,y \in R^{-1}(D_0), x\neq y$ : It suffices to assume $x,y\in \phi _k[0,x_r]$ , for some k, since $\phi _j\{0,x_r\}\subseteq D_0$ for all j. R is a bijection on this interval, which implies $R(x) \neq R(y)$ . We obtain

$$ \begin{align*} \frac{|R(x)-R(y)|}{|x-y|} &= R'(c)\leq R'(0) & \text{for some } c\in \phi_k[0,x_r]. \end{align*} $$

Thus,

$$ \begin{align*} |x-y|&\geq \frac{|R(x)-R(y)|}{R'(0)}\geq \frac{C_0}{R'(0)} &\text{since } R(x), R(y) \in D_0. \end{align*} $$

The general case follows by induction and this completes the proof.

Theorem 3.5 Suppose there is $D_0$ , so that $\sigma (\Delta _n) \subseteq D_n$ for all n and $R,D_n$ satisfy the conditions of Lemma 3.4. Then,

$$\begin{align*}\inf\{|\lambda-\lambda'|:\lambda\neq\lambda' \text{ are Dirichlet eigenvalues of } \Delta\}>0. \end{align*}$$

Although the following examples are elementary, they are useful for understanding the results obtained in this section and how they apply to other complex scenarios.

Alonso Ruiz shows in [Reference Alonso Ruiz1, Theorem 3] that

$$\begin{align*}\min\{|\lambda - \lambda'|: \lambda \neq \lambda', \text{Dirichlet eigenvalues of } \Delta\} = \lambda_0^{(5)} - \lambda_0^{(2)}>0\end{align*}$$

and

$$\begin{align*}\min\{|\lambda - \lambda'|: \lambda \neq \lambda', \text{Neumann eigenvalues of } \Delta\} = \lambda_0^{(6)}> 0.\end{align*}$$

Thus, the infimum spacing between eigenvalues coincides with the spectral gap, which is positive.

Since the spectrum of the Laplacian on I is explicit, a direct computation shows that the infimum spacing is again the spectral gap.

Note that the Dirichlet spectrum is given by $\{\pi ^2k^2\}_{k=1}^\infty $ . Hence, the spacing between successive eigenvalues is given by

$$\begin{align*}\pi^2(k+1)^2 -\pi^2k^2 =\pi^2((k+1)^2-k^2) =\pi^2(2k+1),\end{align*}$$

which is increasing in k; hence, the infimum spacing is again the spectral gap, which occurs when $k=1$ . Thus,

$$\begin{align*}\min\{|\lambda - \lambda'|: \text{ Dirichlet eigenvalues of } \Delta\} = 3\pi^2> 0.\end{align*}$$

Recall that the Neumann spectrum is given by $\{\pi ^2k^2\}_{k=0}^\infty $ . Thus, the spacing between successive eigenvalues is again given by $\pi ^2(2k+1)$ , which coincides with the Dirichlet case, with minimum occurring when $k=0$ . Thus,

$$\begin{align*}\min \{|\lambda - \lambda'|: \text{Neumann eigenvalues of } \Delta\}= \pi^2>0. \end{align*}$$