Vibration modes of 3n-gaskets and other fractals

Vibration modes of 3n-gaskets and other fractals
Vibration modes of 3n-gaskets and other fractals
N Bajorin, T Chen, A Dagan, C Emmons, M Hussein,
M Khalil, P Mody, B Steinhurst, A Teplyaev
E-mail: [email protected]
Department of Mathematics, University of Connecticut, Storrs CT 06269 USA
Abstract. We study eigenvalues and eigenfunctions (vibration modes) on the
class of self-similar symmetric finitely ramified fractals which includes 3n-gaskets.
We consider such examples as the Sierpinski gasket, a non-p.c.f. analog of the
Sierpinski gasket, the level-3 Sierpinski gasket, a fractal 3-tree, the hexagasket,
and one dimensional fractals. We develop a theoretical matrix analysis, including
analysis of singularities, which allows us to compute eigenvalues, eigenfunctions
and their multiplicities exactly. We support our theoretical analysis by symbolic
and numerical computations.
AMS classification scheme numbers: 28A80, 31C25, 34B45, 60J45, 94C99
PACS numbers: 02.30.Sa, 02.20.Bb, 02.50.Ga, 02.60.Lj, 02.70.Hm
1. Introduction
In this paper we study eigenvalues and eigenfunctions (vibration modes) on the class
of self-similar fully symmetric finitely ramified fractals. Such studies originated in
[40, 41], where it was observed that on the Sierpiński there are highly localized
eigenfunctions corresponding to eigenvalues of very high multiplicity. Later the
spectrum of the Laplacian on the Sierpiński gasket was studied in detail in [13], and an
example of the modified Koch curve was studied in [34, 33]. The main purpose of our
paper is to develop a theoretical matrix analysis, including analysis of singularities,
which allows the computation of eigenvalues, eigenfunctions and their multiplicities
for a large class of more complicated fractals.
Our analysis, in particular, allows the computation of the spectral zeta function
on fractals (see [8, 49]) and the limiting distribution of eigenvalues (i.e. integrated
density of states). The latter is a pure point measure, except in the examples which
are based on the one dimensional interval. This support has a representation
[
supp(κ) = JR D,
where JR is the Julia set of a rational function, which we compute, and D is a
possibly empty set of isolated points (if D is infinite, it accumulates to JR ). Also, our
analysis allows the computation of eigenvalues and eigenfunctions by a highly accurate
hierarchical iterative procedure, which does not involve large matrix calculations‡ and
is illustrated in Figures 1, 2 and 3.
‡ see http://www.math.uconn.edu/~teplyaev/fractals/
Vibration modes of 3n-gaskets and other fractals
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Figure 1. A basic Neumann eigenfunction on the Sierpiński gasket, three
dimensional views.
There is a large body of physics and mathematics literature devoted to analysis
on fractals. A small sample of it, containing many references, is [2, 5, 14, 44] and
[23, 24, 25, 26, 43, 45, 46, 47, 48, 50]. In particular, tools for the numerical analysis
of the Sierpiński gasket were developed in [7, 16], and fractal antenae were considered
in [12, 21, 37, 39].
Our study is closely related to the analysis of self-similar graphs [27, 28, 29, 35,
36, 42, and references therein], quantum graphs [30, 31, and references therein], selfsimilar groups [4, 17, 18, 19, 38, 51, and references therein], and the relation between
electrical circuits and Markov chains [6, 10, 11, and references therein].
This paper is organized as follows. In Section 2 we give the definition of the finitely
ramified fractals with full symmetry, on which the graphs which we consider are based.
In Section 3 we introduce spectral self-similarity, Schur complement and a Drichlet-toNeumann map, and show how the resolvent of the Laplacian can be computed by an
iterative procedure. In Section 4 we analyze the singularities of our map and obtain
general formulas for eigenvalues and their multiplicites. We also obtain formulas for
corresponding eigenprojectors. In the subsequent sections we use our general method
to analyze the following examples: the Sierpiński gasket (Section 5), a non-p.c.f. analog
of the Sierpiński gasket (Section 6), the level-3 Sierpiński gasket (Section 7), a fractal
3-tree (Section 8), the hexagasket (Section 9), the unit interval as a self-similar set
Vibration modes of 3n-gaskets and other fractals
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Figure 2. A basic Neumann eigenfunction on the level-3 Sierpiński gasket, three
dimensional views.
(Section 10), and the diamond fractal (Section 11).
2. Finitely ramified fractals with full symmetry.
A compact connected metric space F is called a finitely ramified self-similar set if
there are injective contraction maps
ψ1 , ..., ψm : F → F
such that
F =
m
[
i=1
ψi (F )
Vibration modes of 3n-gaskets and other fractals
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Figure 3. A Neumann eigenfunction on the level-3 Sierpiński gasket, three
dimensional views.
and for any n and for any two distinct words w, w′ ∈ Wn = {1, ..., m}n we have
Fw ∩ Fw ′ = V w ∩ V w ′ ,
where Fw = ψw (F ) and Vw = ψw (V0 ). It is assumed that V0 is a finite set of at
least two points, which often is called the boundary of F . Here for a finite word
w = w1 ...wn ∈ Wn we denote
ψw = ψw1 ◦ ... ◦ ψwn .
We define
Vn =
m
[
ψi (Vn−1 ) =
i=1
[
Vw
w∈Wn
and call this set the vertices of level or depth n.
There is a natural infinite self-similar sequence of “fractal” finite graphs Gn with
vertex set Vn defined as follows. For each n > 0 and w ∈ Wn we define Gw as a
complete graph with vertices Vw . Then, by definition,
[
Gw .
Gn =
w∈Wn
Note that Gn has no loops, but is allowed to have multiple edges, depending on the
structure of the fractal F , as in Section 6. The degree of a vertex x in graph Gn is
denoted by degn (x). The degrees of vertices are uniformly bounded in all our examples
except the non-p.c.f. analog of the Sierpiński gasket in Section 6.
Vibration modes of 3n-gaskets and other fractals
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The main object of our study are eigenvalues and eigenfunctions on the
probabilistic graph Laplacians ∆n on Gn , which are defined by
∆n f (x) = f (x) −
1
degn (x)
X
f (y)
(x,y)∈E(Gn )
where E(Gn ) denotes the set of edges of the graph Gn . For convenience we denote
the matrix of ∆n by Mn in the standard basis of functions on Vn .
Our main geometric assumption is that for any permutation σ : V0 → V0 there is
an isometry gσ : F → F that maps any x ∈ V0 into σ(x) and preserves the self-similar
structure of F . This means that there is a map geσ : W1 → W1 such that
ψi ◦ gσ = gσ ◦ ψgeσ (i)
for all i ∈ W1 . The group of isometries gσ is denoted by G.
It is well know that the eigenvalues and eigenfunctions of ∆n describe vibration
modes of so called cable systems modeled on the graph Gn . They are also can
be considered as discrete approximations to eigenvalues and eigenfunctions of a
continuous self-similar Laplacian ∆µ on F . This continuous self-adjoint Laplacian
is the generator of a self-similar diffusion process on F which can be defined in the
standard way in terms of a self-similar resistance (Dirichlet) form on F , that is for
any f in a suitably defined domain Dom∆µ of the Neumann Laplacian we have
Z
E(f, f ) =
f ∆µ f dµ
F
where µ is the standard suitably normalized self-similar (Hausdorff, Bernoulli) measure
on F .
A G-invariant resistance form E on F is self-similar with energy renormalization
factor ρ if for any f ∈ Dom(E) we have
E(f, f ) = ρ
m
X
E(fi , fi ).
i=1
Here we use the notation fw = f ◦ ψw for any w ∈ Wn . Such resistance forms in the
case of p.c.f. fractals were studied in detail in [23]. The finitely ramified case can be
studied in a similar way because of the general results in [24]. In particular, existence
and uniqueness, up to a scalar multiplier, of the local regular self-similar G-invariant
resistance form E is shown in [50]. Moreover, one can show that
E = lim ρ−n En
n→∞
where the usual graph energy is
En (f, f ) =
X
(x,y)∈E(Gn )
2
f (x) − f (y)
and that
(ρm)−n ∆n f (x) −−−−−→ ∆µ f (x)
n→∞
Vibration modes of 3n-gaskets and other fractals
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for any function f for which ∆µ f ∈ C(F ) and any x ∈ V∗ = ∪n>0 V0 . In addition, one
has a relation
d
R(0) > 1
ρm =
dz
where R(z) is the rational function that appears in the spectral decimation process,
and is one of the most important objects in our study.
The standard and almost trivial example of the self-similar energy and Laplacian
in a finitely ramified situation is the case of F = [0, 1]. In this case we can take m = 2
with ψ1 (x) = 21 x and ψ2 (x) = 12 x + 21 , the self-similar measure µ is the usual Lebesgue
measure, ∆µ f = −f ′′ and
E(f, f ) =
Z
1
′
2
(f (x)) dx =
0
Z
0
1
′′
−f f dx =
Z
f ∆µ f dµ
F
for any f ∈ Dom(∆µ ) = {f : f ′ ∈ L2 [0, 1], f ′ (0) = f ′ (1) = 0}. Then we of course have
ρ = 2 and
4−n ∆n f (x) =
2f (x) − f (x − 21n ) − f (x +
4n
1
2n )
−−−−−→ −f ′′ (x)
n→∞
for any f ∈ C 2 [0, 1]. The cases F = [0, 1] with m = 3 and m = 4 are discussed in
Section 10.
Although in general the fractal F is an abstract metric space, in our examples
F ⊂ R2 and the metric on F is the restriction of the usual Euclidean metric in R2 .
Moreover, the isometries gσ are restrictions of isometries of R2 that maps F into itself
and preserves the self-similar structure of F . We do not require that contractions ψi
are similitudes (see Section 6). One can easily construct more involved and higher
dimensional examples for which our methods apply.
3. Spectral self-similarity, Schur complement and Drichlet-to-Neumann
map
If we have a matrix M given in a block form
A B
M=
C D
(1)
then its Schur complement is
A − BD−1 C.
(2)
In our work one of the most important objects is the Schur complement of the matrix
M − z which is defined by
S(z) = A − z − B(D − z)−1 C.
(3)
Note that we use a convention that M − z denotes M − zI where I is the identity
matrix of the same size as M . Similarly, A − z and D − z denote the matrices A and
D minus z times the identity matrix of the appropriate size.
Our interest in S(z) can be explained as follows. As the initial step in our
calculations, we would like to relate the eigenvalues and eigenvectors of the larger
Laplacian matrix M = M1 and the eigenvalues and eigenvectors of a smaller Laplacian
Vibration modes of 3n-gaskets and other fractals
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matrix M0 . In our setup, the blocks A and D in (1) correspond to outer (boundary)
and interior vertices respectively.
Suppose v is an eigenvector of M which is partitioned into its boundary part v0
and interior part v1′ . Then eigenvalue equation
M v = zv
can be written as
A
C
B
D
v0
v1′
=z
v0
v1′
(4)
or as two equations
Av0 + Bv1′
Cv0 + Dv1′
= zv0
= zv1′
(5)
/ σ(D),
From the second equation we obtain v1′ = −(D − z)−1 Cv0 , provided z ∈
which implies
S(z)v0 = 0.
(6)
If v0 is also an eigenvector of M0 with an eigenvalue z0 , then we would like to
relate (6) with
(M0 − z0 )v0 = 0.
(7)
According to [47, 36], we can write z0 = R(z) if we solve what is our main equation
S(z) = φ(z) M0 − R(z) ,
(8)
where φ(z) and R(z) are scalar (meaning not matrix-valued) rational functions.
Proposition 3.1. For a given fully symmetric self-similar structure on a finitely
ramified fractal F there is a unique rational function φ(z) and R(z) that solve equation
(8).
Proof. Clearly S(z) is a matrix valued rational function. By our main symmetry
assumption in the previous section, for any z the matrix S(z) is a linear combination
of the identity matrix and M0 , which implies the proposition.
Remark 3.2. From the calculations above one can see that S(λ) is the so called
Drichlet-to-Neumann map for the Laplacian ∆1 .
In our examples M0 is a matrix that has 1 on the diagonal and − N01−1 off the
diagonal. Therefore we have that
φ(z) = −(N0 − 1)S1,2 (z)
and
R(z) = 1 −
S1,1 (z)
.
φ(z)
Here N0 is the number of boundary vertices, which is the number of points in V0 .
From the calculations above we have the following theorem.
Vibration modes of 3n-gaskets and other fractals
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Theorem 3.1. Suppose that z is not an eigenvalue of D, and not a zero of φ. Then
z is an eigenvalue of M with an eigenvector
v if and only if R(z) is an eigenvalue of
v0
M0 with an eigenvector v0 , and v =
where
v′
v ′ = −(D − z)−1 Cv0 .
This implies, in particular, that there is an one-to-one map from the eigenspace of M0
corresponding to R(z) onto the eigenspace of M corresponding to z
v0 7→ v = T (z)v0
where
T (z) = I0 − (D − z)−1 C.
Naturally, the map v0 7→ v is called the eigenfunction extension map, and T (z) is
called the eigenfunction extension matrix.
The theorem above suggest the following definition of the so called exceptional
set
E(M0 , M ) = σ(D) ∪ {z : φ(z) = 0}.
Once we have computed the functions R(z) and φ(z) using the smaller matrices
M0 and M = M1 , we can compute the spectrum of much larger matrices Mn by
induction using the following results.
We use notation
An Bn
Mn =
Cn Dn
for the block decomposition of Mn corresponding to the representation
[
Vn = Vn−1 Vn′
where Vn′ = Vn \Vn−1 .
Theorem 3.2. For all n > 0 we have a relation
∗
Pn−1 (Mn − z)−1 Pn−1
=
1
(Mn−1 − R(z))−1 ,
φ(z)
where Pn−1 is defined as the restriction operator from Vn to Vn−1 . We often identify
Pn−1 with the orthogonal projection from ℓ2 (Vn ) onto the subspace of functions with
support in Vn−1 .
Suppose that zn ∈
/ E(M0 , M ). Then zn is an eigenvalue of Mn with an eigenvector
vn if and only if
zn−1 = R(zn )
vn−1
where
is an eigenvalue of Mn−1 with an eigenvector vn−1 , and vn =
vn′
vn′ = −(Dn − zn )−1 Cn vn−1 .
In such a situation vn′ is called the continuation of the eigenfunction vn−1 from Vn−1
to Vn \Vn−1 .
Vibration modes of 3n-gaskets and other fractals
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One can obtain information about the extension of eigenfunctions and
eigenprojectors from Vn−1 to Vn by the following theorem.
Theorem 3.3. Let Pn,zn be the eigenprojector of Mn corresponding to an eigenvalue
zn ∈
/ E(M0 , M ), and Pn−1,zn−1 be the eigenprojector of Mn−1 corresponding to
eigenvalue zn−1 = R(zn ). Then
Pn,zn =
1
d
φ(zn ) dz
R(zn )
Tn (zn ) Pn−1,zn−1 (Pn−1 − Bn (Dn − zn )−1 Pn′ )
(9)
where
Tn (z) = (Pn−1 − (Dn − z)−1 Cn )
and Pn′ is defined as the restriction operator from Vn to Vn \Vn−1 . We often identify Pn′
with the orthogonal projection from ℓ2 (Vn ) onto the subspace of functions that vanish
on Vn−1 . In this case Pn′ = In − Pn−1 .
Proof. First we will prove the key formula for the proof of these theorems. This
formula is not related to spectral similarity and is a known fact. Essentially, it shows
how to find the inverse of a matrix given in a two-by-two block form. To simplify
notation we assume that n = 1 and M1 = M .
Suppose that matrices D − x and A − x − B(D − x)−1 C are invertible. Then
M − x is invertible and
(M − x)−1 = (D − x)−1 +
+ (P0 − (D − x)−1 C)(A − x − B(D − x)−1 C)−1 (P0 − B(D − x)−1 )
(10)
It is enough to prove this formula for x = 0, i.e. to prove
M −1 = D−1 + (P0 − D−1 C)(A − BD−1 C)−1 (P0 − BD−1 )
(11)
−1
provided that D and A − BD C are invertible.
We have
M D−1 = (P1′ + P0 )M D−1 P1′ = P1′ + P0 M D−1 P1′
and
P0 M (P0 − D−1 C) = M P0 − P1′ M P0 − P0 M D−1 C = P0 (A − BD−1 C).
Thus
M (D−1 P1′ + (P0 − D−1 C)(A − BD−1 C)−1 (P0 − BD−1 P1′ )) =
= P1′ + P0 M D−1 P1′ + P0 (P0 − BD−1 P1′ ) = P1′ + P0 = I.
That is what (11) says.
To obtain the proof Theorem 3.2, note that (10) implies
(M − x)−1 = (D − x)−1 P1′ +
+ (P0 − (D − x)−1 C)(φ(x)M0 − φ1 (x))−1 (P0 − B(D − x)−1 P1′ ),
(12)
where φ1 (z) = φ(z)R(z). The statements of Theorem 3.3 follow if we use the standard
spectral representation
X
M=
zPz .
z∈σ(M )
and pass to the limit as x → z in this formula.
Vibration modes of 3n-gaskets and other fractals
10
Remark 3.3. For n = 1 these theorems are also true for the adjacency matrix graph
Laplacian. For n > 1 it is important that we consider probabilistic graph Laplacian, or
a multiple of it. For instance, [9, 46] and related works usually consider the Laplacian,
∆n , multiplied by 4.
4. Analysis of the exceptional values.
It is not enough to restrict ourself to values of z outside of the exceptional set
E(M0 , M ). In fact, this set is very interesting because it often contains eigenvalues of
high multiplicity, which in turn often correspond to localized eigenfunctions.
We first formulate a proposition that gives the multiplicities of such eigenvalues,
and is used extensively to analyze examples in the rest of the paper. Then we prove
a theorem which implies the proposition.
We write multn (z) for the multiplicity of z as an eigenvalue of Mn . By definition,
multn (z) = 0 if z is not an eigenvalue. Notation dimn is used for the dimension of
ℓ2 (Vn ) which is the same as the number of points in Vn .
Proposition 4.1. (i) If z ∈
/ E(M0 , M ), then
multn (z) = multn−1 (R(z)),
(13)
and every corresponding eigenfunction at depth n is an extension of an
eigenfunction at depth n − 1.
(ii) If z ∈
/ σ(D), φ(z) = 0 and R(z) has a removable singularity at z, then
multn (z) = dimn−1 ,
(14)
and every corresponding eigenfunction at depth n is localized.
(iii) If z ∈ σ(D), both φ(z) and φ1 (z) have poles at z, R(z) has a removable singularity
d
at z, and dz
R(z) 6= 0, then
multn (z) = mn−1 multD (z) − dimn−1 + multn−1 (R(z)),
(15)
and every corresponding eigenfunction at depth n vanishes on Vn−1 .
(iv) If z ∈ σ(D), but φ(z) and φ1 (z) do not have poles at z, and φ(z) 6= 0, then
multn (z) = mn−1 multD (z) + multn−1 (R(z)).
(16)
In this case mn−1 multD (z) linearly independent eigenfunctions are localized,
and multn−1 (R(z)) more linearly independent eigenfunctions are extensions of
corresponding eigenfunction at depth n − 1.
(v) If z ∈ σ(D), but φ(z) and φ1 (z) do not have poles at z, and φ(z) = 0, then
multn (z) = mn−1 multD (z) + multn−1 (R(z)) + dimn−1
(17)
provided R(z) has a removable singularity at z.
In this case there are
mn−1 multD (z)+dimn−1 localized and multn−1 (R(z)) non-localized corresponding
eigenfunctions at depth n.
Vibration modes of 3n-gaskets and other fractals
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(vi) If z ∈ σ(D), both φ(z) and φ1 (z) have poles at z, R(z) has a removable singularity
d
R(z) = 0, then
at z, and dz
multn (z) = multn−1 (R(z)),
(18)
provided there are no corresponding eigenfunctions at depth n that vanish on Vn−1 .
In general we have
multn (z) = mn−1 multD (z) − dimn−1 + 2multn−1 (R(z))
(19)
(vii) If z ∈
/ σ(D), φ(z) = 0 and R(z) has a pole z, then multn (z) = 0 and z is not an
eigenvalue.
(viii) If z ∈ σ(D), but φ(z) and φ1 (z) do not have poles at z, φ(z) = 0, and R(z) has
a pole z, then
multn (z) = mn−1 multD (z)
(20)
and every corresponding eigenfunction at depth n vanishes on Vn−1 .
In the next theorem we establish the relation between eigenprojectors of spectrally
similar operators. Namely, we show how one can find the eigenprojector Pn,z
of Mn corresponding to an eigenvalue z, if the eigenprojector Pn−1,R(z) of Mn−1
corresponding to eigenvalue R(z) is known.
We state this theorem for n = 1 and M = M1 , and the analogous relation holds
for any n > 1. As before, we define φ1 (z) = φ(z)R(z).
Theorem 4.1. (i) In the case of Proposition 4.1(i),
Pz =
1
d
φ(z) dz
R(z)
(P0 − (D − z)−1 C)P0,R(z) (P0 − B(D − z)−1 ).
(21)
(ii) In the case of Proposition 4.1(ii),
Pz = (P0 − (D − z)−1 C)(ψ0 (z)M0 − ψ1 (z))−1 (P0 − B(D − z)−1 )
(22)
where ψ0 (x) = φ(x)/(z − x) and ψ1 (x) = φ1 (x)/(z − x). This implies, in
particular, that there is an one-to-one map v0 7→ v = v0 − (D − z)−1 Cv0 from
ℓ2 (V0 ) onto the eigenspace of M corresponding to z.
(iii) In the case of Proposition 4.1(iii), the poles of φ(z) and φ1 are simple and so
R(z) has a removable singularity at z, Pz PD,z = Pz and P0 Pz = 0, which means
that the corresponding eigenfunctions of M vanish on V0 .
Moreover,
rankPD,z − rankPz = rank(ψ0 (z)M0 − ψ1 (z)I0 ) = corankP0,R(z)
where ψ0 (x) = φ(x)(z − x) and ψ1 (x) = φ1 (x)(z − x).
In addition, the following relations hold
Pz = PD,z +
1
PD,z C(M0 − R(z))−1 (I0 − P0,R(z) )BPD,z
ψ0 (z)
(23)
and PD,z CP0,R(z) = 0. Note that I0 − P0,R(z) is the projector from ℓ2 (V0 ) onto
the space, where (D − z)−1 is a well defined bounded operator.
Vibration modes of 3n-gaskets and other fractals
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(iv) In the case of Proposition 4.1(iv),
Pz = PD,z +
1
d
φ(z) dz
R(z)
(P0 − (D − z)−1 C)P0,R(z) (P0 − B(D − z)−1 )
(24)
and the projector PD,z is orthogonal to the second term in the right hand side of
this formula. In particular, Pz PD,z = PD,z .
(v) In the case of Proposition 4.1(v), Pz is the sum of the right hand sides in (22)
and (24).
(vi) In the case of Proposition 4.1(vi), provided there are no corresponding
eigenfunction at depth n that vanish on Vn−1 , we have
Pz =
2
d2
ψ(z) dz
2 R(z)
(P0 − (D − z)−1 C)P0,R(z) (P0 − B(D − z)−1 ).
(25)
In general, this formula is combined with 23.
(vii) In the case of Proposition 4.1(vii) we formally have Pz = 0.
(viii) In the case of Proposition 4.1(viii) we have Pz = PD,z .
Proof. Item (i) is the same as Theorem 3.3; it is inserted here also for the sake of
completeness.
To prove item (ii), we pass to the limit as x → z in the formula 12, which can be
re-written as
(M − x)−1 = (D − x)−1 +
1
(P0 − (D − x)−1 C)(ψ0 (x)M0 − ψ1 (x))−1 (P0 − B(D − x)−1 ). (26)
+
z−x
Then the statements to be proved follow if we pass to the limit as x → z in this
formula.
To prove item (iii), we again pass to the limit as x → z in formula (12). We see
that P0 Pz 6= 0 if and only if
lim (x − z)2 (ψ0 (x)M0 − ψ1 (x)I0 )−1 6= 0,
x→z
that is only possible if
follows from (12).
Note that
d
dz R(z)
= 0. Therefore P0 Pz = 0 in our case. Relation (23)
ψ0 (z)M0 − ψ1 (z)I0 = −P0 M PD,z M P0
if z ∈ σ(D). Hence rank(ψ0 (z)M0 − ψ1 (z)I0 ) = rank(PD,z − Pz ). In addition, we
have that ψ0 (z)M0 − ψ1 (z)I0 is nonpositive.
Also we see that P0 (M − z)−1 P0 is a bounded operator on ℓ2 (V0 ) and so we have
P0 (M − z)−1 P0 = lim (z − x)(ψ0 (x)M0 − ψ1 (x)I0 )−1 . Hence P0 (M − z)−1 P0 = 0 if
x→z
and only if R(z) has a pole at z or R(z) ∈ ρ(M0 ). If R(z) has a removable singularity
at z then
d
0
ψ0 (z) R(z)P0 (M − z)−1 P0 = PR(z)
.
dz
To prove item (iv), note that the relation Pz PD,z = PD,z easily follows from the
fact that φ and φ1 do not have poles. Then, if we restrict everything to the orthogonal
complement of the image of PD,z , we can apply item (i) of this theorem.
Item (v) follows from items (ii) and (iv). The proof of item (vi) is a combination
of the proofs of items (i) and (iii). Items (vii) and (viii) easily follow from (12).
Vibration modes of 3n-gaskets and other fractals
13
5. Sierpiński gasket.
Spectral analysis on the Sierpiński gasket originates from physics papers [40, 41] and
is well known [7, 13, 46, 47]. In this section we show how one can study it using our
methods. Note that recently Sierpiński lattices appeared as the Schreier graphs of so
called Hanoi towers groups [19, 38, 51].
x2
T
T
T x4
x6 T
T
T T
T
T
x1
x5
x3
.......
.............
.................
..... .....
.............. ...............
........................................
..........
.......
.....................
.....................
......... ......... .......... ..........
.
... ... ... .... ... ... ... ...
...........................................................................................
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.....................
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..... .....
..... .....
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............ .............
.................................................
................................................
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......
.
.
.
.
.
.
.
.
... ...
.... ...
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................ .................. .................. .................. .................. .................. .................. ..................
.
.
.
.
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.
.
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..
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.
.
.
.
.....
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.............. ..............
........................................
........................................
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.
.
.
......................................
.
.
.
........
.........
.........
.........
.........
........
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........
....................
.....................
.....................
.....................
.....................
....................
.....................
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..... .... ..... .... ..... .... ..... .... ..... .... ..... .... ..... .... ..... ....
....... ........ ....... ....... ....... ....... ....... ....... ....... ....... ....... ........ ....... ....... ....... .......
...........................................................................................................................................................................................................................................................................................................................................................
Figure 4. The Sierpiński gasket and its V1 network.
Figure 4 shows the depth one approximation to the Sierpiński gasket. The depth
1 Laplacian matrix M = M1 , which is obtained from the above figure, is


1
0
0
0
− 21 − 21
 0
1
0
− 21 0
− 21 


1
1
 0
0
1
−2 −2 0 

M =
1
1
1
1 
 0
 1 − 4 − 14 1 1 − 4 − 41 
 −
0
−4 −4 1
−4 
4
1
1
1
1
−4 −4 1
−4 −4 0
The eigenfunction extension map is

1

(D − z)−1 C = 
−5+2(7−4z)z
2(−1+z)
5+2z(−7+4z)
2(−1+z)
5+2z(−7+4z)
From these we have that
φ(z) =
and
2(−1+z)
5+2z(−7+4z)
1
−5+2(7−4z)z
2(−1+z)
5+2z(−7+4z)
2(−1+z)
5+2z(−7+4z)
2(−1+z)
5+2z(−7+4z)
1
−5+2(7−4z)z



3 − 2z
5 − 14z + 4z 2
R(z) = (5 − 4z)z.
The eigenvalues of M written with multiplicities are
3 3 3 3 3
, , , , ,0
σ(M ) =
2 2 2 4 4
and the corresponding eigenvectors are {-1, -1, 0, 0, 0, 1}, {-1, 0, -1, 0, 1, 0}, {0, -1,
-1, 1, 0, 0}, {2, 0, -2, -1, 0, 1}, {2, -2, 0, -1, 1, 0}, {1, 1, 1, 1, 1, 1}. The eigenvalues
of D written with multiplicities are
5 5 1
σ(D) =
, ,
4 4 2
and the corresponding eigenvectors are {-1, 0, 1}, {-1, 1, 0}, {1, 1, 1}. The equation
ϕ = 0 has as its solution { 32 } so the exceptional set is
5 1 3
E(M0 , M ) =
.
, ,
4 2 2
Vibration modes of 3n-gaskets and other fractals
14
2
1.75
1.5
1.25
1
0.75
0.5
0.25
1
0.5
1.5
2
Figure 5. The graph of R(z) for the Sierpiński gasket.
We can find the multiplicities of these exceptional values by using Proposition 4.1.
For the value 54 , which is a pole of φ(z) and in σ(D), we use Proposition 4.1(iii)
to find the multiplicities:
mult1 ( 54 ) = 2 − 3 + 1 = 0,
mult2 ( 54 ) = 6 − 6 + 1 = 1,
mult3 ( 54 ) = 18 − 15 + 1 = 4,
For the value 12 , which is also a pole of φ(z) and in σ(D), we again use
Proposition 4.1(iii) to find the multiplicities:
mult1 ( 12 ) = 1 − 3 + 2 = 0,
mult2 ( 12 ) = 3 − 6 + 3 = 0,
mult3 ( 12 ) = 9 − 15 + 6 = 0,
/ σ(D) and φ(z) = 0, we use Proposition 4.1(ii) to find
For the value 32 , since 32 ∈
the multiplicities. Here the multiplicity of 23 in the nth depth is equal to the dimension
at depth n − 1.
mult1 ( 32 ) = 3,
mult2 ( 32 ) = 6,
mult3 ( 32 ) = 15,
Table 1 shows the ancestor-offspring structure of the eigenvalues of the Sierpiński
gasket. The symbol * indicates branches
√
5 − 25 − 16z
ξ1 (z) =
8
and
ξ2 (z) =
5+
√
25 − 16z
8
Vibration modes of 3n-gaskets and other fractals
15
of the inverse function R−1 (z) computed at the ancestor value z. By Proposition 4.1(i)
the ancestor and the offspring have the same multiplicity. The empty columns
represent exceptional values. If they are eigenvalues of the appropriate Mn , then
the multiplicity is shown in the right hand part of the same row.
z ∈ σ(M0 )
0
3
2
mult0 (z)
1
2
z ∈ σ(M1 )
0
mult1 (z)
1
z ∈ σ(M2 )
0
mult2 (z)
1
z ∈ σ(M3 )
0
mult3 (z)
1
5
4
3
4
1
2
3
2
2
5
4
3
3
4
ξ1 ( 43 ) ξ2 ( 43 )
2
5
4
2
1
2
3
∗
∗
∗
∗
∗ ∗
3
4
2
2
2
2
3 3
6
3
2
5
4
6
1
1
2
∗ ∗
3
2
5
4
1 1 15 4
Table 1. Ancestor-offspring structure of the eigenvalues on the Sierpiński gasket
By induction one can obtain the following proposition, which is known in the case
of the Sierpiński gasket (see [13, 46, 47]).
Notation R−n A is used for the preimage of a set A under the n-th composition
power of the function R.
Proposition 5.1. (i) σ(M0 ) = {0, 32 }.
(ii) For any n > 0
σ(Mn ) ⊂
and for any n > 1 we have
σ(Mn ) = { 23 }
n
[
m=0
[
R−m {0, 32 }
n−1
[
m=0
!
R−m {0, 43 } .
In particular, for n > 2
σ(Mn ) = {0,
3
2}
[
n+1
n−1
[
m=0
(iii) For any n > 0, dimn = 3 2 +3 .
(iv) For any n > 0, multn (0) = 1.
n
(v) For any n > 0, multn ( 23 ) = 3 2+3 .
!
R−m { 34 }
[
n−2
[
m=0
!
R−m { 54 }
.
Vibration modes of 3n-gaskets and other fractals
16
3n−k−1 +3
for n > 1, 0 6 k 6 n − 1.
2
3n−k−1 −1
=
for n > 2, 0 6 k 6 n − 2.
2
(vi) If z ∈ R−k { 43 } then multn (z) =
(vii) If z ∈ R−k { 54 } then multn (z)
Corollary 5.2. The normalized limiting distribution of eigenvalues (the integrated
density of states) is a pure point measure κ with the the set of atoms
!
!
∞
∞
[ [
[ [
3
5
3
R−m { 4 }
R−m { 4 } .
{2}
m=0
Moreover,
m=0
1
κ { 32 } = ,
3
and
κ({z}) = 3−m−1
if z ∈ R−m { 34 , 45 }.
For the Sierpiński gasket we also demonstrate how one can compute the
eigenprojectors for the two most interesting eigenvalues, z = 23 and z = 45 . For
the former case we use Theorem 4.1(ii). We compute ψ0 ( 32 ) = 1 and ψ1 ( 32 ) = − 23 and
so
−1 −1 −1 Pn+1, 23 = Pn − Dn − 32
.
(27)
Cn Mn + 23
Pn − Bn Dn − 32
For the case z =
5
4
we use Theorem 4.1(iii) with R( 54 ) = 0 and ψ0 ( 45 ) =
Pn+1, 54 = PDn , 54 + 12 PDn , 45 Cn Mn−1 Bn PDn , 54 .
1
12
and so
(28)
Note that one can show that the term In − Pn,0 is the projector to the orthogonal
complement to constants and so can be omitted in this case. Note also that PDn , 45
has a simple block structure with blocks


2 −1 −1
1
−1 2 −1 
3
−1 −1 2
and that Dn has a block structure with blocks


4 −1 −1
1
−1 4 −1  .
4
−1 −1 4
The matrices of Cn and Bn also have similarly simple block structure with block
equivalent, depending on the labeling of vertices, to


0 −1 −1
1
−1 0 −1  .
4
−1 −1 0
except for boundary vertices.
The computation of the eigenprojectors using
Theorem 4.1(i) plays an important role in [47].
Vibration modes of 3n-gaskets and other fractals
17
6. A non-p.c.f. analog of the Sierpiński gasket.
Several non-p.c.f. analogs of the Sierpiński gasket were introduced in [50]. Here we
analyze the simplest one of them. This fractal can be constructed as a self-affine
fractal in R2 using 6 affine contractions, as shown in [50]. It is finitely ramified but
not p.c.f. in the sense of Kigami. Figure 6 shows the V1 network for this fractal.
....
.....
.........
..........
....... .........
.. .
...... .......
........ ..........
........... ..........
................. ...................
.
......... .........
.............. ...............
.......................... ..........................
........................................... ............................................
.
.
.
... .... ....... ...
..... ...................... .........
...
......
....
................. ........................ ...................
.
.
.
.
.
.
.
........................................................ .......................................................
.................................................................... ..............................................................................
.
.
.
.
.
. . ..
......................... ...................
................... ......................
...................................................................................................................................................................................................................................................
.
.
.
.
.
.
.
.............................. ............
... .. ...
............................
........... ............................
..............................
...................
...............
..................
..
............
x2
x4
x7
x6
x5
x1
x3
Figure 6. The non-p.c.f. analog of the Sierpiński gasket and its V1 network.
The matrix of the depth-1

1
 0

 0

M =
 01
 −
 41
 −
4
− 61
Laplacian M1 = M is
0
1
0
− 41
0
− 14
− 16
0
0
1
− 41
− 41
0
− 61
and the eigenfunction extension map is

1
− 6−18z+12z
2

−5+6z
2)

(D − z)−1 C =  12(1−3z+2z
−5+6z
 12(1−3z+2z
2)
1
−3+6z
0
− 41
− 41
1
0
0
− 16
− 41
0
− 14
0
1
0
− 16
− 14
− 14
0
0
0
1
− 16
−5+6z
12(1−3z+2z 2 )
1
− 6−18z+12z
2
−5+6z
2
12(1−3z+2z )
1
−3+6z
Moreover, we compute that
φ(z) =
and
− 12
− 12
− 12
− 12
− 12
− 12
1










−5+6z
12(1−3z+2z 2 )
−5+6z
12(1−3z+2z 2 )
1
− 6−18z+12z
2
1
−3+6z



.

15 − 14z
24 − 72z + 48z 2
24z(z − 1)(2z − 3)
.
14z − 15
The eigenvalues of D, written with multiplicites, are
3
1
σ(D) =
, 1, 1,
2
2
R(z) = −
with corresponding eigenvectors {-1, -1, -1, 1}, {-1, 0, 1, 0}, {-1, 1, 0, 0}, {1, 1, 1, 1}.
One can also compute
3 3 5 5 3 3
σ(M ) =
, , , , , ,0
2 2 4 4 4 4
Vibration modes of 3n-gaskets and other fractals
18
2
1.75
1.5
1.25
1
0.75
0.5
0.25
1
0.5
1.5
2
Figure 7. The graph of R(z) for the non-p.c.f. analog of the Sierpiński gasket.
with the corresponding eigenvectors {-1, -1, -1, 0, 0, 0, 1}, {-1, -1, -1, 1, 1, 1, 0}, {-1,
0, 1, -1, 0, 1, 0}, {-1, 1, 0, -1, 1, 0, 0}, {1, 0, -1, -1, 0, 1, 0}, {1, -1, 0, -1, 1, 0, 0}, {1,
1, 1, 1, 1, 1, 1}.
It is easy to see that φ(z) = 0 has one solution { 15
14 }. Thus, the exceptional set is
3
1 15
E(M0 , M ) =
.
, 1, ,
2
2 14
z ∈ σ(M0 )
0
3
2
mult0 (z)
1
2
z ∈ σ(M1 )
0
mult1 (z)
1
z ∈ σ(M2 )
0 1
mult2 (z)
1
1
3
2
3
2
1
2
3
4
5
4
3
2
2
2
2
∗ ∗ ∗ ∗ ∗ ∗
1
2
2 2 2 2 2 2
3
4
5
4
1
2
1
3
2
2 2 1 6 7
Table 2. Ancestor-offspring structure of the eigenvalues on the non-p.c.f. analog
of the Sierpiński gasket.
15
To begin the analysis of the exceptional values, note that 14
is a pole of R(z) and
therefore is not an eigenvalue by Proposition 4.1(vii). We are interested in the values
of R(z) in the other exceptional points, which are
R(1) = R( 23 ) = 0
It is easy to see that 1 and
1
2
and R( 21 ) = 32 .
are poles of φ(z) and so we can use Proposition 4.1(iii)
Vibration modes of 3n-gaskets and other fractals
19
to compute the multiplicities. We obtain
mult1 (1) = 2 − 3 + 1 = 0,
mult1 ( 21 ) = 1 − 3 + 2 = 0,
mult2 (1) = 12 − 7 + 1 = 6,
and
mult2 ( 12 ) = 6 − 7 + 2 = 1.
Since 23 is not a pole of φ(z), we can use Proposition 4.1(iv) to compute the
multiplicities
mult1 ( 32 ) = 1 + 1 = 2
and
mult2 ( 32 ) = 6 + 1 = 7.
The ancestor-offspring structure of the eigenvalues on the non-p.c.f. analog of the
Sierpiński gasketis shown in Table 2. The symbol * indicates branches of the inverse
function R−1 (z) computed at the ancestor value. The multiplicity of the ancestor is the
same as that of the offspring by Proposition 4.1(i). The empty columns correspond
to the exceptional values. If they are eigenvalues of the appropriate Mn , then the
multiplicity is shown in the right hand part of the same row.
Sn
Theorem 6.1. (i) For any n > 0 we have that σ(∆n ) ⊂ m=0 R−m ({0, 32 }) and
3 5 3
σ(∆1 ) = {0, 4 , 4 , 2 }.
(ii) For n > 2 we have that
!
!
[ n−1
[
[ n−2
[
3 5
1
3
R−m { 4 , 4 }
R−m { 2 , 1} .
σ(∆n ) = {0, 2 }
m=0
m=0
11 + 4 · 6n
.
5
(iv) For any n > 0 we have multn (0) = 1.
(v) For any n > 1 we have multn ( 23 ) = 6n−1 + 1.
(vi) For any n > 1 and z ∈ R1−n { 34 , 45 } we have that multn (z) = 2.
(vii) For any 0 6 m 6 n − 2 and z ∈ R−m { 43 , 45 } we have that
(iii) For any n > 0 we have dimn =
multn (z) = multn−m−1 ( 23 ) = 6n−m−2 + 1.
11 · 6n−m−2 − 6
.
5
6n−m − 6
(ix) For any 0 6 m 6 n − 2 and z ∈ R−m {1} we have multn (1) =
.
5
(viii) For any 0 6 m 6 n − 2 and z ∈ R−m { 12 } we have multn ( 12 ) =
Proof. For this fractal we have σ(∆0 ) = {0, 32 } with mult0 ( 23 ) = 2 and, for the purposes
of Proposition 4.1, m = 6.
Item (i) is obtained above in this section.
Item (ii) follows from the subsequent items.
Item (iii) is straightforward by induction.
Item (iv) follows from Proposition 4.1(i) because 0 is a fixed point of R(z).
Vibration modes of 3n-gaskets and other fractals
20
Item (v) easily follows from Proposition 4.1(iv).
Items (vi) and (vii) follows from the items above.
Items (viii) and (ix) follows from Proposition 4.1(iii) because
multn ( 12 ) = 6n−1 · 1 −
11 + 4 · 6n−1
11 · 6n−2 − 6
+ 6n−2 + 1 =
,
5
5
multn (1) = 6n−1 · 2 −
11 + 4 · 6n−1
6n − 6
+1=
.
5
5
Corollary 6.1. The normalized limiting distribution of eigenvalues (the integrated
density of states) is a pure point measure κ with the the set of atoms
∞
[
m=0
5
where κ { 32 } =
and
24
5 −m−2
6
4
11 −m−2
6
κ({z}) =
4
1
κ({z}) = 6−m
4
κ({z}) =
R−m { 32 , 1},
if
z ∈ R−m { 43 , 45 };
if
z ∈ R−m { 21 };
if
z ∈ R−m {1}.
7. Level-3 Sierpiński gasket.
The level-3 Sierpiński gasket is shown in Figure 8. It had been used as an example
in several works [3, 20, 46, and references therein]. In particular, the spectrum is
computed in the recent paper [9] independently of our work.
The matrix for the depth-1 Laplacian M1 = M is


1
0
0
− 21 − 12 0
0
0
0
0
 0
1
0
0
0
− 12 − 12 0
0
0 


1
1
 0

−
0
0
1
0
0
0
0
−
2
2
 1

1
1
1 
 −
0
0
1
−
0
0
0
−
−
4
4
4 
 41
 −
0
0
− 41 1
− 14 0
0
0
− 41 
4
.
M =
1
1
1
 0
0
−4 1
−4 0
0
− 41 
−4 0


 0
− 41 0
0
0
− 14 1
− 41 0
− 14 


 0
0
− 14 0
0
0
− 14 1
− 14 − 14 


 0
0
0
− 41 1
− 14 
0
− 14 − 41 0
0
0
0
− 16 − 16 − 16 − 16 − 61 − 16 1
Vibration modes of 3n-gaskets and other fractals
21
and the eigenfunction extension map (D − z)−1 C is
 −24+109z−132z2 +48z3
−9+7z












3(1−6z+4z 2 )(15−32z+16z 2 )
−24+109z−132z 2 +48z 3
3(1−6z+4z 2 )(15−32z+16z 2 )
−4+3z
3(−5+34z−4z 2 +16z 3 )
−9+7z
3(1−6z+4z 2 )(15−32z+16z 2 )
−9+7z
3(1−6z+4z 2 )(15−32z+16z 2 )
−4+3z
3(−5+34z−4z 2 +16z 3 )
1
− 3−18z+12z
2
3(1−6z+4z 2 )(15−32z+16z 2 )
−4+3z
3(−5+34z−4z 2 +16z 3 )
−24+109z−132z 2 +48z 3
3(1−6z+4z 2 )(15−32z+16z 2 )
−24+109z−132z 2 +48z 3
3(1−6z+4z 2 )(15−32z+16z 2 )
−4+3z
3(−5+34z−4z 2 +16z 3 )
−9+7z
3(1−6z+4z 2 )(15−32z+16z 2 )
1
− 3−18z+12z2
−4+3z
3(−5+34z−4z 2 +16z 3 )
−9+7z
3(1−6z+4z 2 )(15−32z+16z 2 )
−9+7z
3(1−6z+4z 2 )(15−32z+16z 2 )
−4+3z
3(−5+34z−4z 2 +16z 3 )
−24+109z−132z 2 +48z 3
3(1−6z+4z 2 )(15−32z+16z 2 )
−24+109z−132z 2 +48z 3
3(1−6z+4z 2 )(15−32z+16z 2 )
1
− 3−18z+12z
2
Moreover, we compute that
φ(z) =
and
(2z − 3)(6z − 7)
3(4z − 5)(4z − 3)(1 − 6z + 4z 2 )
R(z) =
6z(z − 1)(4z − 5)(4z − 3)
.
6z − 7
x2
T
T
Tx5
x7 T
T
T T
T x8
x4 x10T
T
T
T
T T T
T
T
T
x3
x1
x6
x9
....
..........
................
......................................
....... ........ ....
........................................................
....................
..........
.
.
..
..... .....
...............
...................................... ...................................
....... ....... ...... ........ ....... ........
............................................................................................................
..........
....
.....
............................... .................................... ...................................
.
.
..
. ..
..
.
............................ ............................ ................................
..............................................................................................................................
....... ...... .. ... .. ....... ...... .. ...... ....... ....................
..................
............
...... ........
.....................................
.................................................
..
.
.
............ ............ ............
.... .. ....
.......................................................
...............................................
.
..
.
..........
..
............
...........
................................ .....................................
........................................... ..............................................
.
.
..
..
.
.
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........... ........... ............ ......... ........... ...........
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.
.
..
.
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..
.
.
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.
..
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.
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..... . ................ ............. ..........
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........
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......
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.
..
.
..... ... .... ..... ..... .... .... ..... ..... ... .... ..... ..... ... .... ..... ..... .... .... ..... ..... ... .... ..... .... ... ....
...........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
Figure 8. The level-3 Sierpiński gasketand its V1 network.
2
1.75
1.5
1.25
1
0.75
0.5
0.25
0.5
1
1.5
2
Figure 9. The graph of R(z) for the level-3 Sierpiński gasket.
The eigenvalues of D, written with multiplicities are
√ 5 5 3 3 1
√
3 1
, (3 + 5), , , , , (3 − 5)
σ(D) =
2 4
4 4 4 4 4







.





Vibration modes of 3n-gaskets and other fractals
22
One can also compute
√ 1
√
√ 1
√
3 3 3 3 1
1
, , , , (3 + 2), (3 + 2), 1, (3 − 2), (3 − 2), 0
σ(M ) =
2 2 2 2 4
4
4
4
We find that φ(z) = 0 has two solutions { 76 }, { 23 }. Thus, the exceptional set is
√ 5 3 1
√ 7
3 1
E(M0 , M ) =
.
, (3 + 5), , , (3 − 5),
2 4
4 4 4
6
z ∈ σ(M0 )
0
3
2
mult0 (z)
1
2
z ∈ σ(M1 )
0
1
mult1 (z)
1
1
z ∈ σ(M2 ) 0 1
mult2 (z)
11
3 5
4 4
√
3± 2
4
3 5
4 4
2
√
3± 5
4
3
2
2
4
∗∗∗ ∗
∗∗ ∗∗∗ ∗∗∗
√
√
3± 2 3± 5
4
4
1111
22222222
4 4
3
2
3 4
4 5
16 3 3
Table 3. Ancestor-offspring structure of the eigenvalues on the level-3 Sierpiński
gasket.
To begin the analysis of the exceptional values, note that find√the poles of R(z)
√
and see if it is an exceptional value It is easy to see that 34 , 54 , 14 (3 − 5) and 14 (3 + 5)
are poles of φ(z) and so we can use Proposition 4.1(iii) to compute the multiplicities.
We obtain
mult1 ( 34 ) = 2 − 3 + 1 = 0,
mult2 ( 43 ) = 12 − 10 + 1 = 3,
mult1 ( 45 ) = 2 − 3 + 1 = 0,
mult2 ( 45 ) = 12 − 10 + 1 = 3,
√
mult1 ( 3±4 5 ) = 1 − 3 + 2 = 0,
√
mult2 ( 3±4 5 ) = 6 − 10 + 4 = 0.
√
Note that R( 34 ) = R( 45 ) = 0 and R( 3±4 5 ) = 32 . Also, 23 is not a pole of φ(z) but
φ( 32 ) = 0 and therefore we use Proposition 4.1(v) to compute the multiplicities. We
obtain
mult1 ( 32 ) = 1 + 0 + 3 = 4,
mult2 ( 32 ) = 6 + 0 + 10 = 16.
Vibration modes of 3n-gaskets and other fractals
23
The ancestor-offspring structure of the eigenvalues on the level-3 Sierpiński gasket
is shown in Table 3. The multiplicity of the ancestor is the same as that of the offspring
by Proposition 4.1(i). The empty columns correspond to the exceptional values. If
they are eigenvalues of the appropriate Mn , then the multiplicity is shown in the right
hand part of the same row.
Sn
Theorem 7.1. (i) For any n > 0 we have that σ(∆n ) ⊂ m=0 R−m ({0, 32 }) and
√
σ(∆1 ) = { 23 , 41 (3 ± 2), 54 , 43 }.
(ii) For n > 0 we have that
√ !! [ 3
[
3± 5
.
R−(n−1)
σ(∆n ) = (R−n (0))
4
2
(iii) For n > 0 we have dimn = 3 + 57 (6n − 1).
(iv) For n > 0 we have that multn (0) = multn (1) = 1.
(v) For n > 2 and for z ∈ R−k (1), 0 > k 6 2 we have that multn (z) = 1.
2 · 6n + 8
(vi) For n > 0 we have that multn ( 23 ) =
.
5
(vii) For n > 2 and 0 6 k 6 n − 2 we have for z ∈ R−k { 34 , 45 } that
multn (z) = 35 (6n−k−1 − 1).
Note as a special case k = 0 which gives the multiplicities of
3
4
and 54 .
√
(viii) For n > 1 with 0 6 k 6 n − 1 we have that for z ∈ R−k ( 3±4 2 )
multn (z) = multn−k−1 ( 32 ) =
2 · 6n−k−1 + 8
.
5
√
(ix) For any n > 1 with 0 6 k 6 n − 1 we have that for z ∈ R−k ( 3±4 5 )
multn (z) = 0.
Proof. For this fractal we have σ(∆0 ) = {0, 32 } with mult0 ( 23 ) = 2 and, for the purposes
of Proposition 4.1, m = 6.
Item (i) is obtained above in this section.
Item (ii) follows from the subsequent items.
Item (iii) is straightforward by induction.
Item (iv) follows from Proposition 4.1(i) because 0 is a fixed point of R(z) and
because R(1) = 0.
Item (v) easily follows from Proposition 4.1(i) and Item (iv).
Item (vi) follows from the previous items and Proposition 4.1(v).
Item (vii) follows from Proposition 4.1(iii).
Item (viii) follows from Proposition 4.1(i).
Item (ix) follows from Proposition 4.1(iii), and as a consequence none of these
values appear in the spectrum.
Vibration modes of 3n-gaskets and other fractals
24
Corollary 7.1. The normalized limiting distribution of eigenvalues (the integrated
density of states) is a pure point measure κ with the the set of atoms
!
[ [
∞
√
3
3 5 3± 2
R−m { 4 , 4 , 4 } .
2
m=0
2
Moreover, κ { 23 } = and
7
κ({z}) = 37 6−m−1
if
κ({z}) = 27 6−m−1
if
z ∈ R−m { 43 , 45 };
n √ o
z ∈ R−m 3±4 2 .
8. A fractal 3-tree.
The fractal tree is a fractal that is approximated by triangles as shown in Figure 10,
but in the limit is a topological tree. It appeared as the limit set of the Gupta-Sidki
group, see [4, 38, and references therein].
..
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x
x
x
x
x
x
x6
Figure 10. The fractal 3-tree and its V1 network.
The depth-1 Laplacian matrix M1 = M is

1
0
0
− 21 0
 0
1
0
0
− 12

 0
1
0
0
 1 0
1
 −
0
0
1
−
4
 4
1
1

−4 1
−4 0
M = 0
 0
− 14 − 41 − 14
 1 0
 −
− 21 0
 2 01 0
 0
−2 0
0
− 12
1
0
0
−2 0
0
0
0
− 12
− 14
− 14
1
0
0
− 12
− 12
0
0
− 14
0
0
1
0
0
0
− 12
0
0
− 14
0
0
1
0
0
0
− 12
0
0
− 14
0
0
1














x3
Vibration modes of 3n-gaskets and other fractals
25
and the eigenfunction extension map (D − z)−1 C is

5+2z(−7+4z)
2(−1+z)










9−8z(6+z(−9+4z))
2(−1+z)
(−3+4z)(3+4z(−3+2z))
2(−1+z)
(−3+4z)(3+4z(−3+2z))
−7+8(3−2z)z
(−3+4z)(3+4z(−3+2z))
1
9−8z(6+z(−9+4z))
1
9−8z(6+z(−9+4z))
(−3+4z)(3+4z(−3+2z))
5+2z(−7+4z)
9−8z(6+z(−9+4z))
2(−1+z)
(−3+4z)(3+4z(−3+2z))
1
9−8z(6+z(−9+4z))
−7+8(3−2z)z
(−3+4z)(3+4z(−3+2z))
1
9−8z(6+z(−9+4z))
2(−1+z)
(−3+4z)(3+4z(−3+2z))
2(−1+z)
(−3+4z)(3+4z(−3+2z))
5+2z(−7+4z)
9−8z(6+z(−9+4z))
1
9−8z(6+z(−9+4z))
1
9−8z(6+z(−9+4z))
−7+8(3−2z)z
(−3+4z)(3+4z(−3+2z))
From here, we compute that
φ(z) =






.




3 − 2z
9 − 48z + 72z 2 − 32z 3
and
R(z) = 4z(z − 1)(4z − 3).
2
1.75
1.5
1.25
1
0.75
0.5
0.25
0.5
1
1.5
2
Figure 11. The graph of R(z) for the fractal tree.
The eigenvalues of D written with multiplicities are
√ 3 3 1
√ 3 3 1
, ,
σ(D) =
3+ 3 , , ,
3− 3
2 2 4
4 4 4
and √ the √corresponding
eigenvectors are {1, 0, −1, −1, 0, 1}, {1, −1, 0, −1, 1, 0},
√
1− 3 1− 3 1− 3
{ 2 , 2 , 2 , 1, 1, 1}, {− 12 , 0, 21 , −1, 0, 1}, {− 12 , 21 , 0, −1, 1, 0}, and
√
√
√
{ 1+2 3 , 1+2 3 , 1+2 3 , 1, 1, 1}.
Computing the eigenvalues of M with multiplicities gives
1 1
3 3 3 3 3
, , , , , 1, , , 0
σ(M ) =
2 2 2 2 2
4 4
and the corresponding eigenvectors are {0, 0, −1, 0, 0, 0, 0, 0, 1}, {0, −1, 0, 0, 0, 0,
0, 1, 0}, {−1, 0, 0, 0, 0, 0, 1, 0, 0}, {1, 0, −1, −1, 0, 1, 0, 0, 0}, {1, −1, 0, −1, 1, 0, 0,
Vibration modes of 3n-gaskets and other fractals
26
0, 0}, {1, 1, 1, −1, −1, −1, 1, 1, 1}, {−1, 0, 1, − 21 , 0, 12 , −1, 0, 1}, {−1, 1, 0, − 21 , 12 ,
0, −1, 1, 0}, {1, 1, 1, 1, 1, 1, 1, 1, 1}.
The only solution of φ(z) = 0 is 32 . As such, the exceptional set is
√ 1
√
3 3 1
E(M0 , M ) =
, , (3 + 3), (3 − 3) .
2 4 4
4
For analysis of exceptional values, one can find R(z) at each exceptional point by
√ 1
√
3
1 1
R−1 (0) = 0, , 1
and R−1 ( 23 ) =
, (3 − 3), (3 + 3) .
4
4 4
4
Using Proposition 4.1, one can determine the multiplicities of the exceptional values.
For the value 32 , which is a zero of φ(z), we use Proposition 4.1(v) to find the
multiplicities.
mult1 ( 23 ) = 40 (2) + 0 + 3 = 5,
mult2 ( 23 ) = 41 (2) + 0 + 9 = 17.
For the value
multiplicities.
3
4,
which is a pole of φ(z), we use Proposition 4.1(iii) to find the
mult1 ( 43 ) = 40 (2) − 3 + 1 = 0,
mult2 ( 43 ) = 41 (2) − 9 + 1 = 0.
√
√
For the values 14 (3 + 3) and 14 (3 − 3), which are poles of φ(z), we use
Proposition 4.1(iii) to find the multiplicities.
√
mult1 ( 14 (3 ± 3)) = 40 (1) − 3 + 2 = 0,
√
mult2 ( 41 (3 ± 3)) = 41 (1) − 9 + 5 = 0.
z ∈ σ(M0 )
0
3
2
mult0 (z)
1
2
z ∈ σ(M1 )
0
mult1 (z)
1
z ∈ σ(M2 )
0
mult2 (z)
1
3
4
3
4
1
1
4
1
2
√
3± 3
4
3
2
5
1
∗ ∗ ∗ ∗ ∗ ∗
1
4
1
1 1 1 2 2 2
5
√
3± 3
4
3
2
17
Table 4. Ancestor-offspring structure of the eigenvalues of the fractal tree.
The ancestor-offspring structure of the eigenvalues of the Fractal Tree is shown
in Table 4. As before, the symbol * indicates branches of the inverse function R−1 (z)
computed at the ancestor value. The multiplicity of the ancestor equals that of the
offspring by Proposition 4.1(i). The exceptional values are represented by the empty
columns. If they are eigenvalues of the appropriate Mn , then the multiplicity is shown
in the right hand part of the same row.
Vibration modes of 3n-gaskets and other fractals
27
Theorem 8.1. (i) For any n > 0 we have that σ(∆n ) ⊂
√
and σ(∆1 ) = { 23 , 41 (3 ± 3), 34 }.
(ii) For n > 2 we have that
3 [
σ(∆n ) = 0,
2
n−1
[
k=0
R−k
Sn
m=0
R−m ({0, 23 })
S
{ 32 }
!
1
,1
.
4
And for n = 1 we have σ(∆1 ) = {0, 41 , 1, 32 }.
(iii) For n > 0 we have dimn = 3 + 2(4n − 1).
(iv) For n > 0 we have multn (0) = multn (1) = 1.
(v) For n > 2 with 0 6 k 6 n − 2 we have that if z ∈ R−k (1) then
multn (z) = multn−k (1) = 1.
(vi) For n > 0 we have that
multn ( 32 ) = 4n + 1.
(vii) For n > 1 with 0 6 k 6 n we have for z ∈ R−k ( 14 ) that
multn (z) = multn−k ( 14 ) = multn−k−1 ( 32 ) = 4n−k−1 + 1.
(viii) For n > 1 we have multn ( 34 ) = 0.
√
(ix) For n > 1 with 0 6 k 6 n we have that if z ∈ R−k ( 3±4 3 ) then multn (z) = 0.
Proof. For this fractal we have σ(∆0 ) = {0, 32 } with mult0 ( 23 ) = 2 and, for the purposes
of Proposition 4.1, m = 6.
Item (i) is obtained above in this section.
Item (ii) follows from the subsequent items.
Item (iii) is straightforward by induction.
Item (iv) follows from Proposition 4.1(i) because 0 is a fixed point of R(z) and
because R(1) = 0.
Item (v) easily follows from Proposition 4.1(i) and Item (iv).
Item (vi) follows from the previous items and Proposition 4.1(v).
Item (vii) follows from Proposition 4.1(i).
Items (viii) and (ix) follow from Proposition 4.1(iii), and as a consequence none
of these values appear in the spectrum.
Corollary 8.1. The normalized limiting distribution of eigenvalues (the integrated
density of states) is a pure point measure κ with the the set of atoms
!
[ [
∞
1
3
.
R−m
2
4
m=0
Moreover, κ { 23 } = 21 , and κ({z}) = 12 4−m−1 if z ∈ R−m { 14 }.
Vibration modes of 3n-gaskets and other fractals
x2
T
T
x10 Tx8
x4
x5
T
T
T T Tx7
x11T
T
T
T
T
x9
Tx3
x1 x12 T
T
T T
x6
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28
Figure 12. The hexagasket and its V1 network.
9. Hexagasket.
The hexagasket, or the hexakun, is a fractal which in different situations [1, 6, 23,
46, 50, 52, 53, and references therein] is called a polygasket, a 6-gasket, or a (2, 2,
2)-gasket. The depth-1 approximation to it is shown in Figure 12.
The matrix of the depth-1 Laplacian M1 = M is


1
0
0
− 12 − 12 0
0
0
0
0
0
0
 0
0
0
0
0 
1
0
0
0
− 12 − 12 0


1
1
 0
1
0
0
0
0
−2 −2 0
0
0 

 1 0
 −
0
1
− 14 0
0
0
− 41 − 14 0
0 

 14 0
 −
− 14 1
− 14 0
0
0
0
− 14 0 

 4 01 0
 0
−4 0
0
− 14 1
− 14 0
0
0
− 14 0 

M =
 0
− 41 0
0
0
− 14 1
− 14 0
0
0
− 14 


 0
0
0
− 14 1
− 41 0
0
− 14 
0
− 41 0


 0
0
− 41 − 14 0
0
0
− 14 1
− 14 0
0 


 0
0
0
0
− 21 1
0
0 
0
0
− 12 0


 0
0
0
0
− 12 − 12 0
0
0
0
1
0 
0
0
0
0
0
0
− 12 − 12 0
0
0
1
and the eigenfunction extension map (D − z)−1 C is

−4+z(23+4z(−9+4z))
2 )(7+8z(−3+2z))
 (1−6z+4z
−4+z(23+4z(−9+4z))

 (1−6z+4z2 )(7+8z(−3+2z))

−2+(7−4z)z
 (1−6z+4z2 )(7+8z(−3+2z))

−1+z

 (1−6z+4z2 )(7+8z(−3+2z))

 (1−6z+4z2−1+z
)(7+8z(−3+2z))

−2+(7−4z)z

2
 (1−6z+4z )(7+8z(−3+2z))

−3+4(3−2z)z
 (1−6z+4z
2 )(7+8z(−3+2z))

−3+4(3−2z)z

 (1−6z+4z2 )(7+8z(−3+2z))
1
− (1−6z+4z2 )(7+8z(−3+2z))
Moreover, we compute that
φ(z) =
−1+z
(1−6z+4z 2 )(7+8z(−3+2z))
−2+(7−4z)z
(1−6z+4z 2 )(7+8z(−3+2z))
−4+z(23+4z(−9+4z))
(1−6z+4z 2 )(7+8z(−3+2z))
−4+z(23+4z(−9+4z))
(1−6z+4z 2 )(7+8z(−3+2z))
−2+(7−4z)z
(1−6z+4z 2 )(7+8z(−3+2z))
−1+z
(1−6z+4z 2 )(7+8z(−3+2z))
1
− (1−6z+4z2 )(7+8z(−3+2z))
−3+4(3−2z)z
(1−6z+4z 2 )(7+8z(−3+2z))
−3+4(3−2z)z
(1−6z+4z 2 )(7+8z(−3+2z))

−2+(7−4z)z
(1−6z+4z 2 )(7+8z(−3+2z))

−1+z

(1−6z+4z 2 )(7+8z(−3+2z)) 

−1+z
(1−6z+4z 2 )(7+8z(−3+2z)) 

−2+(7−4z)z

(1−6z+4z 2 )(7+8z(−3+2z)) 

−4+z(23+4z(−9+4z))
(1−6z+4z 2 )(7+8z(−3+2z)) 

−4+z(23+4z(−9+4z))

(1−6z+4z 2 )(7+8z(−3+2z)) 

−3+4(3−2z)z
(1−6z+4z 2 )(7+8z(−3+2z)) 

1

− (1−6z+4z2 )(7+8z(−3+2z))

−3+4(3−2z)z
(1−6z+4z 2 )(7+8z(−3+2z))
3 + 4(z − 2)z
(4z 2 + 6z − 1) (7 + 8z(2z − 3))
Vibration modes of 3n-gaskets and other fractals
and
R(z) =
29
2z(z − 1)(7 − 24z + 16z 2 )
.
2z − 1
2
1.75
1.5
1.25
1
0.75
0.5
0.25
0.5
1
1.5
2
Figure 13. The graph of R(z) for the hexagasket.
The eigenvalues of D, written with multiplicities, are
√ 1
√ 1
√ 3 3 3 1
.
3± 5 ,
3± 2 ,
3± 2
, , ,
σ(D) =
2 2 2 4
4
4
One can also compute
σ(M ) =
3 3 1 1
3 3 3 3 3 3
, , , , , , 1, , , , , 0
2 2 2 2 2 2
4 4 4 4
with the corresponding eigenvectors {0, 1, 0, 0, 0, 0, −1, 0, 0, 0, 0, 1}, {1, 0, 0, 0, −1,
0, 0, 0, 0, 0, 1, 0}, {1, 0, 0, −1, 0, 0, 0, 0, 0, 1, 0, 0}, {1, 0, −1, −1, 0, 0, 0, 0, 1, 0, 0,
0}, {0, 1, −1, 0, 0, 0, −1, 1, 0, 0, 0, 0}, {1, −1, 0, 0, −1, 1, 0, 0, 0, 0, 0, 0}, {−1, −1,
−1, 0, 0, 0, 0, 0, 0, 1, 1, 1}, {1, −1, 0, 0, 21 , − 21 , 0, 12 , − 21 , −1, 0, 1}, {0, −1, 1, − 12 ,
1 1
1 1
1
1
1 1
1
2 , 0, − 2 , 2 , 0, −1, 1, 0}, {−1, 1, 0, −1, − 2 , 2 , 1, 2 , − 2 , −1, 0, 1}, {0, 1, −1, − 2 , 2 ,
1
1
1, 2 , − 2 , −1, −1, 1, 0}, {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}.
It is easy to see that φ(z) = 0 has two solution 21 and 23 . Thus, the exceptional
set is
√ 1
√ 1
3 1
3± 5 ,
3± 2 ,
.
,
E(M0 , M ) =
2 4
4
2
To begin the analysis of the exceptional values, note that 12 is the pole of R(z)
and therefore is not an eigenvalue
√ by Proposition
√ 4.1(vii).
It is easy to see that 41 (3 ± 2) and 14 3 ± 5 are the four poles of φ(z) and so
we can use Proposition 4.1(iii) to compute the multiplicities. We obtain
√
mult1 ( 41 (3 ± 2)) = 60 · 2 − 3 + 1 = 0,
√
mult2 ( 41 (3 ± 2)) = 61 · 2 − 12 + 1 = 1,
√
mult1 ( 14 (3 ± 5)) = 60 · 1 − 3 + 2 = 0,
√
mult2 ( 41 (3 ± 5)) = 61 · 1 − 12 + 6 = 0.
Vibration modes of 3n-gaskets and other fractals
30
z ∈ σ(M0 )
0
3
2
mult0 (z)
1
2
z ∈ σ(M1 )
0
1
mult1 (z)
1
1
z ∈ σ(M2 ) 0 1
mult2 (z)
11
√
3± 2
4
√
3± 2
4
1
4
3
4
2
2
√
3± 5
4
3
2
6
∗∗ ∗∗
∗∗∗∗ ∗∗∗ ∗
1 3
4 4
1111
22222222
6 6
√
3± 5
4
3
2
√
3± 2
4
30 1 1
Table 5. Ancestor-offspring structure of the eigenvalues on the hexagasket.
The exceptional value 23 is in the spectrum σ(D), not a pole of φ(z) and φ( 32 ) = 0.
For this reason we can use Proposition 4.1(v) to compute the multiplicities.
mult1 ( 32 ) = 60 · 3 + 0 + 3 = 6,
mult2 ( 32 ) = 61 · 3 + 0 + 12 = 30.
As in the other sections, the multiplicities of all eigenvalues at depths 0, 1 and 2 are
shown in Table 5.
Theorem 9.1. (i) σ(M0 ) = {0, 23 }.
3
1 3
and for n > 2 we have
(ii) We have that σ(M1 ) = 0, , , 1,
4 4
2
(
√ )!
! [ n−2
n−1
[
3 [ [
1 3
3± 2
σ(Mn ) = 0,
.
R−m 1, ,
R−m
2
4 4
4
m=0
m=0
(iii) For any n > 0 we have dimn =
6 + 9 · 6n
.
5
6 + 4 · 6n
.
5
(v) For any n > 1 and 0 6 k < n − 1 we have that if x ∈ R−k (1) then multn (z) = 1.
(vi) For any n > 1 and 0 6 k < n − 1 we have that if z ∈ R−k { 14 , 43 } then
(iv) For any n > 0, multn (0) = 1 and multn ( 23 ) =
multn (z) =
6 + 4 · 6n−k−1
.
5
√
(vii) For any n > 2 and 0 6 k < n − 2 we have that if z ∈ R−k ( 3±4 2 ) then
multn (z) =
√
6n−k−1 − 1
.
5
(viii) For n > 0 we have multn ( 3±4 5 ) = 0.
Vibration modes of 3n-gaskets and other fractals
31
Proof. For this fractal we have σ(∆0 ) = {0, 32 } with mult0 ( 23 ) = 2 and, for the purposes
of Proposition 4.1, m = 6.
Item (i) is obtained above in this section.
Item (ii) follows from the subsequent items.
Item (iii) is straightforward by induction.
Item (iv) follows from Proposition 4.1(i) because 0 is a fixed point of R(z), and
from Proposition 4.1(v).
Items (v) and (vi) follow from Proposition 4.1(i).
Items (vii) and (viii) follow from Proposition 4.1(iii).
Corollary 9.1. The normalized limiting distribution of eigenvalues (the integrated
density of states) is a pure point measure κ with the the set of atoms
(
√ )!
[ [
∞
1 3 3± 2
3
.
, ,
R−m
2
4 4
4
m=0
4
Moreover, κ { 23 } = , and
9
κ({z}) = 49 6−m−1
if
κ({z}) = 19 6−m−1
if
z ∈ R−m { 41 , 43 };
√
z ∈ R−m { 3±4 2 }.
10. One dimensional interval as a self-similar set.
In this section we show how our results allow us to recover classically known
information about the spectrum of the discrete Laplacians that approximate the usual
one dimensional continuous Laplacian. The unit interval [0,1] can be represented as
a self-similar set in various ways. Here we consider three cases: when it subdivided
into two, three or four subintervals of equal length. In our notation this means that
m is 2, 3, or 4. The depth-1 networks for these cases are shown in Figure 14. The
first two cases were also discussed in [49]. Note that in each case the function R(z) is
the same as the Chebyshev polynomial of degree m for the interval [0,2], which is the
smallest interval that contains the spectrum of the matrices Mn . It is shown in [49], in
particular, that the iterations of these polynomials are related in a natural way with
the Riemann zeta function.
r
x1
r
x3
r
x1
r
x1
r
x3
r
x3
r
x2
r
x4
r
x4
r
x2
r
x5
r
x2
Figure 14. V1 networks for the interval in cases m = 2, 3, 4 respectively.
Case m = 2. The matrix of the depth-1 Laplacian M1 = M is


1
0
−1
1
−1 
M = 0
1
1
−2 −2 1
Vibration modes of 3n-gaskets and other fractals
32
and the eigenfunction extension map is
(D − z)−1 C =
Moreover, we compute that
φ(z) =
and
1
2(z−1)
1
2(z−1)
.
1
2(1 − z)
R(z) = 2z(2 − z).
The only eigenvalue of D is σ(D) = {1} . One can also compute σ(M ) = {2, 1, 0} with
the corresponding eigenvectors{{-1, -1, 1}, {-1, 1, 0}, {1, 1, 1}}. It is easy to see that
φ(z) 6= 0. Thus, the exceptional set is
E(M0 , M ) = {1} .
To begin the analysis of the exceptional value, note that R(z) does not have
any poles. We are interested in the value of R(z) at the exceptional point, which is
R(1) = 2. It is easy to see that 1 is a pole of φ(z), R(z) has a removable singularity
d
at z, and dz
R(z) = 0. So for all n we can use Proposition 4.1(vi) to compute its
multiplicity
multn (1) = 1.
2
2
2
1.75
1.75
1.75
1.5
1.5
1.5
1.25
1.25
1.25
1
1
1
0.75
0.75
0.75
0.5
0.5
0.5
0.25
0.25
0.25
0.5
1
1.5
2
0.5
1
1.5
2
0.5
1
1.5
2
Figure 15. The graph of R(z) for F = [0, 1] with m = 2, m = 3 and m = 4
respectively.
Case m = 3. The matrix of the depth-1 Laplacian

1
0
−1
 0
1
0
M =
 −1 0
1
2
0
− 12 − 12
and the eigenfunction extension map is
−1
(D − z)
C=
2(z−1)
3−8z+4z 2
1
8z−4z 2 −3
M1 = M is

0
−1 

− 12 
1
1
8z−4z 2 −3
2(z−1)
3−8z+4z 2
Moreover, we compute that
φ(z) =
4 z−
3
2
1
z−
1
2
!
.
Vibration modes of 3n-gaskets and other fractals
33
and
R(z) = z(3 − 2z)2 .
The eigenvalues of D, written with multiplicities, are
3 1
,
σ(D) =
2 2
with corresponding eigenvectors{{-1, 1}, {1, 1}}. One can also compute
3 1
σ(M ) = 2, , , 0
2 2
with the corresponding eigenvectors{{1, -1, -1, 1}, {-2, -2, 1, 1}, {-2, 2, -1, 1}, {1, 1,
1, 1}}. It is easy to see that φ(z) 6= 0. Thus, the exceptional set is
3 1
.
,
E(M0 , M ) =
2 2
Again, note that R(z) does not have any poles. We are interested in the values of
R(z) in the exceptional points, which are
R( 23 ) = 0, R( 21 ) = 2.
Since
d
dz R(z)
= 0 in these points, we can use Proposition 4.1(vi) to obtain
multn ( 23 ) = multn ( 12 ) = 1
for all n.
Case m = 4. The matrix of the depth-1 Laplacian M1 = M is


1
0
−1 0
0
 0
0
0
−1 
 1 1

1

1
−2 0 
M =  −2 0

 0
− 12 
0
− 12 1
0
− 21 0
− 12 1
and the eigenfunction extension map is


(D − z)−1 C = 
3−8z+4z 2
4(−1+5z−6z 2 +2z 3 )
1
− 2−8z+4z
2
1
4(−1+5z−6z 2 +2z 3 )
We compute that
φ(z) =
and
1
4(−1+5z−6z 2 +2z 3 )
1
− 2−8z+4z
2
3−8z+4z 2
4(−1+5z−6z 2 +2z 3 )
1
4 − 20z + 24z 2 − 8z 3
R(z) = 8z(z − 2)(1 − z)2 .
The eigenvalues of D, written with multiplicities, are
√ √ 1
1
2 + 2 , 1,
2− 2
σ(D) =
2
2



Vibration modes of 3n-gaskets and other fractals
34
with corresponding eigenvectors
n √ oo
nn
√ o
.
1, − 2, 1 , {−1, 0, 1}, 1, 2, 1
One can also compute
σ(M ) =
√ √ 1
1
2,
2 + 2 , 1,
2 − 2 ,0 .
2
2
It is easy to see that φ(z) 6= 0. Thus, the exceptional set is
n √ √ o
E(M0 , M ) = 21 2 + 2 , 1, 12 2 − 2 .
To begin the analysis of the exceptional values, note that R(z) does not have any
poles. We are interested in the values of R(z) at the exceptional points, which are
√
√
R( 21 (2 + 2)) = 2, R(1) = 0, R( 21 (2 − 2)) = 2.
Once again,
d
dz R(z)
= 0 at these points, and by Proposition 4.1(vi) we have
multn ( 12 (2 +
√
√
2)) = multn (1) = multn ( 12 (2 − 2)) = 1
for all n.
11. Diamond fractal.
The diamond fractal is shown in figure 16. The diamond self-similar hierarchical lattice
appeared as an example in several physics works, such as [14]. Recently the critical
percolation on the diamond fractal was analyzed in [15].
x3
..............................................
................................................ ...............................................
QQ
................ ............................... ......................................................................................
................................... ...............
................. .........................................
............................... ............................
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
..
..............
..... ......
Q
.. ...
.............
.... .....
...................
..... ....
............
...... .....
..................
...... .....
Q
................
.............
...............
...............
..............
..................
..... .....
...........................
.............
.
.
.
.
.
Q
.
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x2
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x1Q
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x4
Figure 16. The diamond fractal and its V1 network.
We can use the results obtained for the unit interval [0,1] in Section 10, case
m = 2, to develop the spectral decimation method for the diamond fractal. The
matrix of the depth-1 Laplacian M1 = M is


1
0
− 12 − 12
 0
1
− 12 − 12 

M =
 −1 −1 1
0 
2
2
1
− 12 − 12 0
Vibration modes of 3n-gaskets and other fractals
35
and the eigenfunction extension map is now the square matrix with the same entries
1 1
1
(D − z)−1 C = 2(z−1)
1 1
while the functions
φ(z) =
and
1
2(1 − z)
R(z) = 2z(2 − z)
are the same as for the unit interval, σ(D) = {1, 1} has multiplicity two, and
σ(M ) = {2, 1, 1, 0} with the corresponding eigenvectors {-1, -1, 1, 1}, {-1, 1, 0,0},
{ 0,0,-1, 1}, {1, 1, 1,1}. The exceptional set is
E(M0 , M ) = {1} .
Theorem 11.1. (i) For any n > 0 we have that
σ(∆n ) =
n
[
m=0
R−m ({0, 2}).
(ii) For any n > 0 we have dimn = 3 + 2(4n − 1).
(iii) For any n > 0 we have multn (0) = multn (2) = 1.
(iv) For any n > 1 and 0 6 k 6 n − 1 we have multn (z) =
4n−k + 2
if z ∈ R−k (1).
3
Proof. Item (i) follows from (iii) and (iv). Item (ii) is obtained by induction. Item (iii)
follows from Proposition 4.1(i), and the fact that R(0) = R(2) = 0. For the analysis
of the only exceptional value z = 1, note that it is a pole of φ(z), R(1) = 2, R(z) has
d
R(1) = 0. Therefore by Proposition 4.1(vi) we
a removable singularity at 1, and dz
have
n−1
n
multn (1) = 4n−1 · 2 − 2·4 3 +4 + 2 = 4 3+2
for all n > 1. This implies Item (iv).
Corollary 11.1. The normalized limiting distribution of eigenvalues (the integrated
density of states) is a pure point measure κ with the the set of atoms
∞
[
m=0
R−m {1}
and κ ({z}) = 12 4−m if z ∈ R−m {1}.
Acknowledgments
The last author is very grateful to Rostislav Grigorchuk, Volodymyr Nekrashevych,
Peter Kuchment and Robert Strichartz for many useful remarks and suggestions. This
research is supported in part by the NSF grant DMS-0505622.
Vibration modes of 3n-gaskets and other fractals
36
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