# Explicit Spectral Decimation for a Class of Self-Similar Fractals

Authors: Sergio A. Hernández; Federico Menéndez-Conde
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

The method of spectral decimation is applied to an infinite collection of self-similar fractals. The sets considered are a generalization of the Sierpinski Gasket to higher dimensions; they belong to the class of nested fractals and are thus very symmetric. An explicit construction is given to obtain formulas for the eigenvalues of the Laplace operator acting on these fractals.


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## Body

## 1. Introduction

In 1989, J. Kigami [1] gave an analytic definition of a Laplace operator acting on the Sierpinski Gasket; a few years later, this definition was extended to include the Laplacians on a large class of self-similar fractal sets [2], known as postcritically finite sets (PCF sets). The method of spectral decimation introduced by Fukushima and Shima in the 1990s provides a way to evaluate the eigenvalues of Kigami’s Laplacian. In general terms, this method consists in finding the eigenvalues of the self-similar fractal set by taking limits of eigenvalues of discrete Laplacians that act on some graphs that approximate the fractal. The spectral decimation method was applied in [3] to the Sierpinski Gasket, in order to give an explicit construction which allows one to obtain the set of eigenvalues. In [4], it was shown that it is possible to apply the spectral decimation method to a large collection of p.c.f. sets, including the family of fractals known as nested fractals that was introduced by T. Lindstrøm in [5]. In addition to the Sierpinski Gasket, the spectral decimation method has been applied in several specific cases of p.c.f. fractals (e.g., [6–10]); also, the method has been proved useful to study the spectrum of particular fractals that are not p.c.f. (e.g., [11, 12]) and of fractafolds modeled on the Sierpinski Gasket [13]. The spectral decimation method has also shown to be a very useful tool for the analysis of the structure of the spectra of the Laplacians of some fractals (e.g., [14–16]).In the present work, we develop in an explicit way the spectral decimation method for an infinite collection of self-similar sets that we will denote byPn (n≥2 a positive integer). The definition of these sets is given in Definition 1. For the cases n=2,3, they correspond, respectively, to the unit interval and the Sierpinski Gasket. For larger values of n they give a quite natural extension of the Sierpinski Gasket to higher dimensions. The spectral decimation method for the cases n=2,3 is presented with thorough detail in [17]. Our presentation follows this reference to some extent. However, some technical difficulties arise for n≥4. This is mainly due to the fact that—even though the fractals considered are very symmetric—the graphs approximating the fractal are not as homogeneous as the ones approximating the Sierpinski Gasket. For instance, if we consider the graph obtained by taking away the boundary points from Γ1 (see Definition 2 and Figures 1 and 2), then it will be a complete graph only for n≤3. A consequence of this is the appearance of sets of two types of vertices that have to be dealt with separately and which we denote by Fr,s and Gr,s; for n≤3, the sets Gr,s are empty. We also make the observation that the approximating graphs Γk are nonplanar when n>3.Figure 1
The first approximating graphΓ1 for the fractal P4.Figure 2
The graphΓ1 for the fractal P4, minus the boundary points.In Section2, we present general facts about self similar sets, for the sake of completeness and in order to establish notation. In Section 3, we introduce the sets Pn that are the subject of study in this work; at the end of the section, we find the Hausdorff dimension of these fractals when embedded in the Euclidean space. In Section 4, we define the graphs that approximate the self-similar sets Pn and fix more notations. Our main result is presented in Section 5 (Theorem 3); it is shown that the eigenvalues and eigenfunctions of the discrete Laplacians of the approximating graphs can be obtained recursively. Finally, in Section 6, it is shown that the eigenvalues of the Laplace operator in Pn can be recovered by taking limits of the discrete Laplacians; in order to do this, we solve the so-called renormalization problem for this case (see Theorem 5).


## 2. Notation and Preliminaries

We denote bySn the shift space with n symbols. In this work, we will always consider these n symbols to be the numbers 0,1,…,n-1. Sn is a compact space (see, e.g., [18]) when equipped with the metric
(1)δ(a0a1a2a3…;b0b1b2b3…)=rk,0<r<1,
where
(2)k=min{j≥0∣aj≠bj}.
We will use the dot notation a˙, meaning that the symbol a repeats to infinity.Letx=x0x1x2… be an element of Sn  and a∈{0,…,n-1}. We denote by Ta the shift-operator given by
(3)Ta(x)=ax0x1x2….
It is easy to see that
(4)δ(T(x);T(y))=rδ(x;y),
so that Ta is a contraction (by factor r). The space Sn is a self-similar set, equal to n smaller copies of itself, with {T0,…,Tn-1} the corresponding contractions. Even more so, it can be proved (Theorem  1.2.3. in [18]) that if K is any self-similar set, then it is homeomorphic to a quotient space of the form Sn/~ for a suitable equivalence relation.ForK=Sn/~ and a a word of length m(5)a=(a0a1⋯am-1),
denote by Ta the shift operator given by
(6)Ta(x)=a0⋯am-1x0x1x2….
The operator Ta is called an m-contraction, and the sets of the form Ta(K) are known as the cells of levelm of the self-similar set K. We note that, for each choice of m, K is the union of the nm cells of level m.


## 3. The Self-Similar FractalsPn

Here we will introduce the self-similar fractalsPn that are the subject of analysis in this work.Definition 1.
Forn∈ℕ, define Pn as the quotient space Sn/~, with the equivalence relation given by
(7)a0a1a2⋯akbc˙~a0a1a2⋯akcb˙,
for any choice of symbols aj, b, and c.P
1 is a trivial space with only one element, P2 is homeomorphic to a compact interval in ℝ, and P3 is homeomorphic to the well-known Sierpinski Gasket. For any value of n, Pn can be embedded in the Euclidean space; more precisely, there exists a (quite natural) homeomorphism between Pn and a compact self-similar set Kn⊂Rn-1. Below, we define the sets Kn; for these representations of Pn, we will be able to find their Hausdorff dimensions.Taken points x0,…,xn-1∈ℝn-1 that do not lie in the same (n-2)-dimensional hyperplane; for n=3, those points will be the “vertices” of the Sierpinski Gasket. For n=4, the fractal K4 will be some sort of a Sierpinski tetrahedron (see Figure 3), while the four points xj will be the vertices of the tetrahedron.Figure 3
A representation ofK4,  generated with Matlab.Consider the contractions(8)fi(x)=x+xi2,i=1,…,n-1.
We note that fi maps each x∈ℝn-1 to the midpoint of x and xi (hence, leaving xi fixed). Define Kn as the unique compact set such that
(9)Kn=⋃i=0n-1‍fi(Kn).
We note that, for i≠j, the sets fi(Kn) and fj(Kn) intersect at exactly one point: fi(xj)=fj(xi). From this, it follows that the map π:Pn→Kn given by
(10)π(ω0ω1ω2…)=⋂m≥0‍fω0∘fω1∘⋯∘fωm(Kn)
is a well-defined homeomorphism; also, for every k=0,…,n-1, the following diagram commutes (cf. Theorem  1.2.3 in [18]):
(11)The setsKn satisfy the Moran-Hutchinson open set condition namely, there exists a bounded nonempty open set O⊂Pn such that
(12)fi(O)⊂O,∀i∈{0,…,n-1},fi(O)∩fj(O),∀i≠j.
Just take O=Kn∖{x0,…,xn-1}. From this and the fact that Kn is equal to n contractions of itself (by factor 1/2), it follows from Moran’s theorem (Corollary  1.5.9 in [18]) that the Hausdorff dimension of Kn, with respect to theEuclidean metric, is equal to logn/log2.We end this section with two relevant notes.For some values ofn, it might be possible to embed Kn isometrically into the Euclidean space of a dimension m smaller than n-1. Of course, the dimension of the fractal gives a restriction to the minimal value of m.The representationsKn are somehow useful to visualize the self-similar fractals Pn. However, this representation and its metric do not play any role in the analysis carried out in the next sections; we will therefore focus on the more abstract definition of Pn given at the beginning of this section.


## 4. Graph Approximations of Self-Similar Sets

In this and the next sections, we consider the self-similar setPn defined above, for an arbitrary but fixed value of n≥2.LetV0 be the set of points in Pn that have the form k˙ with k=1,…,n-1. We call V0 the boundary of Pn. Likewise, for m∈ℕ, let Vm be the subset of Pn of points of the form a0⋯am-1k˙. In other words, x∈Vm if and only if it belongs to the image of V0 under some m-contraction.Next, we define the graphs that will approximatePn.Definition 2.
Denote byΓ0 the complete graph of n vertices, with V0 being its set of vertices. For m∈ℕ, let Γm be the graph with the set of vertices Vm and edge relation established by requiring x to be connected with y if and only if there exists an m-contraction Ta such that both points x and y are in Ta(V0).We can see that an equivalent formulation is that two verticesx and y share an edge in Γm only when their first m symbols coincide. It is worth noting that even though V0⊂V1⊂V2…, the edge relation is never preserved; this follows from the fact that if x≠y are connected in Γm, then their (m+1)-th symbols cannot be equal, so that they will not be connected in Γm+1.For eachm∈ℕ, let Δm be the graph Laplacian on Γm. We consider the Laplacian as acting on a space with boundary. More precisely, for a real-valued function u defined on Vm and x in Vm∖V0,
(13)Δmu(x)=∑y~x‍(u(x)-u(y)),
with the sum over all vertices y that share an edge with x; the boundary values remain unchanged. Also, u is an eigenfunction of Δ0 with eigenvalue λ, if
(14)Δmu(x)=λu(x),∀x∈Vm∖V0.
We denote by Em(·,·) the associated quadratic form (known as the energy product of the graph):
(15)Em(u,v)=(Δmu,v)=∑x~y‍(u(x)-u(y))(v(x)-v(y)),
for u and v real-valued functions defined on Vm and the sum being taken over the pairs of vertices (x,y) that are connected to each other. Also, we use the abbreviation E(u)=E(u,u).


## 5. Spectral Decimation

Letm>1 and suppose u is an eigenfunction of Δm-1, with eigenvalue λm-1. We will show that it is always possible to extend this function to the domain Vm so that it will be an eigenfunction of Δm (with not the same eigenvalue). In order to do this, we will derive necessary conditions for the extension to be an eigenfunction; in the process, it will become clear that those conditions are also sufficient.Suppose thatu is an eigenfunction of Δm with eigenvalue λm; we aim to write the values of um in Vm∖Vm-1 in terms of its values in Vm-1. Without loss of generality, we can restrict ourselves to the set Vm∩Ta(Pn) for a fixed (m-1)-contraction Ta; this is because the vertices of Γm that belong to the set (Vm∖Vm-1)∩Ta(Pn) are not connected to any vertices outside the cell Ta(Pn). Denote the elements of this set by
(16)xb,c=abc˙,b,c=0,…,n-1.
It is clear that xb,c=xc,b and also that xb,c∈Vm-1 if and only if b=c. This is shown in Figure 4.Figure 4
A cell of levelm-1 of a graph Γm,  approximating P4.For each pointxr,s∈Vm∩Ta(Pn), define the sets of vertices
(17)Fr,s={xi,j∣i≠j,{r,s}∩{i,j}≠∅},Gr,s={xi,j∣i≠j,{r,s}∩{i,j}=∅}.
In other words, Fr,s is the set of vertices (not in Γm-1) that are connected to the vertex xr,s in Γm, and Gr,s is the set of vertices (not in Γm-1, either) that are not connected to it.In the casen=4, the graph Γm∖Γm-1 is an octahedron; hence, for each pair {r,s}, the subgraph determined by the vertices in Fr,s is a 4-cycle one, while Gr,s consists of a single vertex (the one opposite to xr,s in the octahedron). For general Pn, we can see that the following.The graphΓm∖Γm-1 has n(n-1) vertices, all of them with degree 2(n-2). Each of these vertices is connected to another 2 vertices in Γm-1.The subgraph determined byFr,s consists of two complete graphs, each one with n-2 vertices. The two complete graphs are joined to each other pairwise, thus forming a “prism,” (a true prism only in the case n=5, where the base is a 3-cycle one, as shown in Figure 5).The subgraph determined byGr,s has (n-2)(n-3)/2 vertices, each one of them with degree 2(n-4).InΓm, each vertex that belongs to Gr,s is connected to exactly four vertices in Fr,s. On the other hand, each vertex that belongs to Fr,s is connected to n-2 vertices in Gr,s.Figure 5
The graph determined byF0,1 in a graph Γm,  approximating P5.Now, having noted all that, we proceed with the calculations. For everyr≠s we have
(18)(2(n-1)-λm)u(xr,s)=u(xr,r)+u(xs,s)+∑Fr,s‍u(xi,j).Adding this up over all the possible values ofr and s and rearranging terms yields
(19)(2-λm)∑r≠s‍u(xr,s)=(n-1)∑j=0n-1‍u(xj,j),
which for any fixed a≠b can also be written in the form
(20)(2-λm)(u(xa,b)+∑Fa,b‍u(xi,j)+∑Ga,b‍u(xi,j))=(n-1)∑j=0n-1‍u(xj,j).
This, together with (18), allows us to express the sum of the values in Ga,b in terms of u(xa,b) and the values at points in Vm-1; namely, provided λm≠2, we have that
(21)∑Ga,b‍u(xi,j)=u(xa,a)+u(xb,b)-(2n-1-λm)u(xa,b)+(n-1)2-λm∑j=0n-1‍u(xj,j).Next, we will take the sum of the same terms, but only over thexi,j inside the set Fa,b for fixed values a≠b. Since Fa,b contains two complete graphs with n-1 vertices and these complete graphs are pairwise connected to each other, it is clear that each xi,j∈Fa,b is connected to other n-1 vertices in Fa,b. Also, recall that each xi,j∈Ga,b is connected to exactly four vertices in Fa,b. For the vertices in Vm-1, we note that xj,j is connected to n-2 vertices in xi,j∈Fa,b if j=a,b and to only two vertices otherwise.From the preceding discussion, it follows the equality(22)(n-λm)∑Fa,b‍u(xi,j)=4∑Ga,b‍u(xi,j)+(n-2)(u(xa,a)+u(xb,b))+2(n-2)u(xa,b)+2∑j≠a,b‍u(xj,j).Consider the expression given by (18) for {a,b}={r,s}, multiply it by n-λm, and substitute equality (22) into it; this gives after arranging terms
(23)(λm2-(3n-2)λm+2(n2-2n+2))u(xa,b)=4∑Ga,b‍u(xi,j)+2∑j≠a,b‍u(xj,j)+(2(n-1)-λm)(u(xa,a)+u(xb,b)).
We want to get rid of the terms corresponding to Ga,b, so we replace it by (21). After straightforward computations, we can see that for λm≠2(24)(λm2-(3n+2)λm+2n(n+2))u(xa,b)=(λm2-2(n+2)+8n)(u(xa,a)+u(xb,b))+2(2n-λm)∑j≠a,bu(xj,j)(2-λm)-1.
The quadratic equation for λm in the left-hand side has roots n+2 and 2n. The one in the right-hand side has roots 4 and 2n. This gives us the following expression for u(xa,b) in terms of the values of u in Vm-1:
(25)u(xr,s)=(4-λm)(u(xr,r)+u(xs,s))+2∑j≠r,su(xj,j)(2-λm)((n+2)-λm),
valid for any eigenvalue λm≠2,n+2,2n.Forλm=0, this reduces to
(26)u(xr,s)=2n+2(u(xr,r)+u(xs,s))+1n+2∑j≠r,s‍u(xj,j).It is clear from the construction that ifu(xa,b) is defined by (25) and λm is given by (35), then we have that
(27)Δmu(xa,b)=λmu(xa,b)(a≠b).It remains to verify that this is valid as well inVm-1. Of course, this cannot be true for arbitrary values of λm but only at most for specific values depending on λm-1; we will find those values in what follows.Take a point inVm-1, say
(28)xp,p=ap˙,a=a0⋯am-1.
Suppose that ak=q is the last symbol in a that is different from p; we can assume that such symbol exists, since otherwise xp,p would be in the boundary V0. With this, the point xp,p can also be written in the form
(29)xp,p=a′q˙,a′=a0⋯ak-1pq⋯q,
with the necessary number of q’s to make a′ a word of length m-1. Hence, xp,p is in exactly two different (m-1)-cells: Ta(Pn) and Ta′(Pn), corresponding to each one of its two representations.Denote byxr,s′ the points in Ta′(Pn)∩Vm, defined as in (16) for the points in Ta(Pn)∩Vm; in particular, xp,p=xq,q′ (see Figure 6). The value of u in the points xr,s′ is given by the analogue of (25). The vertex xp,p is connected in Γm to the 2(n-1) points of the form xp,j and xq,j′, from which it follows that u is an eigenfunction of Δm with eigenvalue λm, if and only if (25) holds for all xr,s∈Vm∖Vm-1 and the following equality holds for all xp,p∈Vm-1:
(30)(2(n-1)-λm)u(xp,p)=∑i≠p,j≠q‍(u(xp,j)+u(xq,j′)).Figure 6
Two cells of levelm-1 intersecting in a vertex of Γm.On the other hand, since we know thatu is an eigenfunction of Δm-1 with eigenvalue λm-1, we also have that
(31)(2(n-1)-λm-1)u(xp,p)=∑i≠p,j≠q‍(u(xi,i)+u(xj,j′)).Replacing each term in the right-hand side of (30) by its expression given by (25), we can see that
(32)(2(n-1)-λm)u(xp,p)=2(n-1)(4-λm)u(xp,p)+(2n-λm)(2-λm)-1×((n+2)-λm)-1,
and, using (31), this gives
(33)u(xp,p)=[2(n-1)(4-λm)+(2n-λm)(2(n-1)-λm)]×u(xp,p)(2(n-1)-λm)-1(2-λm)-1×((n+2)-λm)-1.Takingu(xp,p)≠0 and cancelling out, after computations, the above equality reduces to the quadratic
(34)λm2-(n+2)λm+λm-1=0,
which in turn gives the following recursive characterization of the eigenvalues:
(35)λm=(n+2)±(n+2)2-4λm-12.Since this procedure can be reversed, we have proved the following result.Theorem 3.
Letλm≠2,n+2, and 2n, and let λm-1 be given by (34). Suppose u is an eigenfunction of Δm-1 with eigenvalue λm-1. Extend u to Vm by (25). Then u is an eigenfunction of Δm with eigenvalue λm. Conversely, if u is an eigenfunction of Δm with eigenvalue λm≠2,n+2, and 2n, then its restriction to Vm-1 is an eigenfunction of Δm-1 with eigenvalue λm-1.


## 6. The Laplacian on the Self-Similar Fractals

In order to define the Laplace operator of a p.c.f. fractal by means of graph approximations, it is required to solve the so-calledrenormalization problem for the fractal (e.g., [17], Chapter 4); roughly, this consists in normalizing the graph energies in Γm in order to obtain a self-similar energy in the fractal by taking the limit. This can be achieved if the energies are such that they remain constant for each harmonic extension from Γm to Γm+1. Below, we do this for the Pn sets.Definition 4.
For a given functionu with domain Vm-1, we call the extension of u to Vm given by (26) its harmonic extension.The next result gives the explicit solution of the renormalization problem forPn.Theorem 5.
Letu:Vm-1→ℝ be arbitrary, and let u′:Vm→ℝ be its harmonic extension. Then
(36)Em(u′)=nn+2Em-1(u).Proof.
Note that the energy at levelk of a given function equals the sum of the energies at all the k′-cells for any k′≤k, since different cells share no edges. This allows us to restrict ourselves to one fixed m-1-cell both while considering Em(u′) and Em-1(u). We use the notation of Section 4 for the vertices of Γm in this cell and write E~ for the energy restricted to this cell. We can readily see that
(37)E~m-1(u)=∑i≠j‍(u(xi,i)-u(xj,j))2.=(n-1)∑i=0n-1‍u2(xi,i)-2∑i≠j‍u(xi,iu(xi,j)).
In order to evaluate the energy E~m, we consider first the edges joining vertices in Vm-1 with vertices in Vm∖Vm-1: the edge joining the vertex xa,a with the vertex xa,k contributes to the energy by
(38)(u(xa,a)-u(xa,k))2=1(n+2)2(nu(xa,a)-2u(xk,k)-∑j≠a,k‍u(xj,j))2.
When adding up this over all possible pairs a≠k, each xr,r will appear n-1 times as the xa,a, another n-1 times as the xk,k, and (n-1)(n-2) as one of the xj,j’s. Each pair double product 2xr,rxs,s will appear twice for {r,s}={a,k},  2(n-2) times for {r,s}={a,j} for some j, also 2(n-2) times for {r,s}={k,j} for some j, and finally (n-2)(n-3) times when both r and s are one of the j’s. All this implies that the contribution to the energy from these edges is, after simplification,
(39)∑a≠k‍(u(xa,a)-u(xa,k))2=(n-1)(n2+n+2)(n+2)2∑i=0n-1‍u2(xi,i)-2(n2+n+2)(n+2)2∑i≠j‍xi,ixj,j.
On the other hand, the contribution from the edge that joins the vertices xa,b and xa,c (in Vm∖Vm-1) equals
(40)(u(xa,b)-u(xa,c))2=1(n+2)2(u(xb,b)-u(xc,c))2.
Taking the sum over all of the vertices in Vm∖Vm-1 yields
(41)∑a≠b≠c‍(u(xa,b)-u(xa,c))2=(n-1)(n-2)(n+2)2∑i=0n-1‍u2(xi,i)-2(n-2)(n+2)2∑i≠j‍u(xi,i)u(xj,j).
From (39) and (41), it follows that the total energy of the cell is
(42)E~m(u′)=n(n-1)n+2∑i=0n-1‍u2(xi,i)-2n(n+2).
This, together with (37), gives
(43)E~m(u′)=nn+2E~m-1(u).
Taking this result for all the m-1-cells concludes the proof.Definition 6.
The energy inPn is given by
(44)E(u)=limm→∞(n+2n)mEm(u).
The domain of E(·) being the space D(n) of functions such that the energy is finite. Write D0(n) for the subspace of D(n) of functions that vanish on the boundary. The energy product E(u,v) can be recovered by the polarization identity.Letμ be a self-similar measure in Pn; the Laplacian Δμ is given by the following:Definition 7 (Kigami’s Laplacian).
Withμ and Δμ as above, one says that u is in the domain of Δμ if there exists a continuous function f such that
(45)E(u,v)=-∫Pn‍fvdμ,∀v∈D0(n).
In such case, we define Δμu=f.Aside from the above weak representation, a pointwise formula can be obtained forΔmu, proceeding exactly in the same way as in [17] (Theorem  2.2.1). In the case where μ is the standard measure in Pn (i.e., the only Borel regular measure such that the measure of every m-cell is equal to n-m), the pointwise formula is
(46)Δμu(x)=n2limm→∞(n+2)mΔmu(x).This leads to the following: if a sequence{λm} is defined recursively by (35) (assuming that λm is never equal to n,n+2, or 2n) and um is given by relation (25), then
(47)λ=n2limm→∞(n+2)mλm
is an eigenvalue of Δμ with eigenfunction u given by the limit um→u. The limit above exists provided that the sign in relation (35) is chosen to be “+” for at most a finite number of times.



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