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

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by  Sergio A. Hernandez

Explicit Spectral Decimation for a Class of
Self–Similar Fractals
2
1 Sergio A. Herna´ndez Federico Men´endez–Conde
0
2

p Centro de Investigaci´on en Matem´aticas
e
Universidad Aut´onoma del Estado de Hidalgo
S
[e-mail]

7 [e-mail]

]
Abstract
P
A The method of spectral decimation is applied to an infinite collection
. of self–similar fractals. The sets considered belong to the class of nested
h
fractals, and are thus very symmetric. An explicit construction is given
t
a to obtain formulas for the eigenvalues of the Laplace operator acting on
m these fractals.
[
In 1989, J. Kigami [5] gave an analytic definition of a Laplace operator act-

1
ing on the Sierpinski Gasket; a few years later, this definition was extended to
v
include Laplacians on a large class of self–similar fractal sets [6], known as post
9
4 critically finite sets (p.c.f. sets). The method of spectral decimation introduced
6 by Fukushima and Shima in the 1990’s provides a way to evaluate the eigen-
1 values of Kigami’s Laplacian. In general terms, this method consists in finding
.
9 the eigenvalues of the self–similar fractal set by taking limits of eigenvalues of
0 discrete Laplacians that act on some graphs that approximate the fractal. The
2 spectraldecimationmethodwasappliedin[1]totheSierpinskiGasket,inorder
1
to give an explicit construction that allowed to obtain the set of eigenvalues. In
:
v [11] it is shown that it is possible to apply the method to a large collection of
Xi p.c.f. sets, which includes the nested fractals defined in [8]. In addition to the
Sierpinski Gasket, several specific cases have been treated in the literature (e.g.
r
a [2, 3, 4, 9, 12]).
In the present work, we develope in an explicit way the spectral decimation
method for an infinite collection of self–similar sets, parametrized by n ≥ 2 (a
positive integer). The definition of these sets is given in Definition 1. The cases
n = 2,3, corresponding respectively to the unit interval and to the Sierpinski
Gasket, have been presented with thorough detail in [10]. 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 con-
sideredherearestillverysymmetric–thegraphsapproximatingthefractalare
not as homogeneous as the ones for the Sierpinski Gasket. For instance, if we
1
Figure 1: The approximating graph Γ to the left (for P ), and Γ without the
1 4 1
boundary to the right.
consider the graph obtained by by taking away the boundary points from Γ
1
(see Definition 2 and Figure 1), then it will be a complete graph only for n≤3;
a consequence of this, is the appereance of sets of vertices that have to be dealt
with (and which we denote by G ) that are not present for n≤3.
r,s
In Section 1, we present general facts about self–similar sets, for the sake of
completeness and in order to establish notation; at the end of the section we
introduce the sets P of particular interest in this work. In Section 2 we define
n
the graphs that approximate the self–similar sets P and fix more notation.
n
Our main result is developed in Section 3 (Theorem 1); it is shown that the
eigenvalues and eigenfunctions of the discrete Laplacians of the approximating
graphs can be obtained recursively. Finally, in Section 4, it is shown that the
eigenvalues of the Laplace operator in P can be recovered by taking limits of
n
thediscreteLaplacians;inordertodothis,wesolvetherenormalizationproblem
for this case (see Theorem 2).
1 Notation and Preliminaries
We denote by S the shift–space with n symbols. In this work we will always
n
consider these n symbols to be the numbers 0,1,...,n−1. S is a compact
n
space (see e.g. [7]) when equipped with the metric
1
δ(a a a a ...;b b b b ...)=
0 1 2 3 0 1 2 3 nk
2
where
k =min{j ≥0 | a (cid:54)=b }.
j j
We will use the dot notation a˙, meaning that the symbol a repeats to infinity.
Let x=x x x ... an element of S , and a∈{0,...,n−1}. We denote by
0 1 2 n
T the shift–operator given by the contraction
a
T (x)=ax x x ...
a 0 1 2
The space S is a self–similar set, equal to n smaller copies of itself, with
n
{T ,...,T } the corresponding contractions. Even more so, it can be proved
0 n−1
(Theorem1.2.3. in[7])thateveryself–similarsetequalsaquotientspaceS /∼
n
for an equivalence relation ∼.
Fora=(a a ...a )aword(inthealphabet{0,...,n−1})oflengthm,
0 1 m−1
we denote by T the shift–operator given by
a
T (x)=a ...a x x x ...
a 0 m−1 0 1 2
and call it an m-contraction; the sets of the form T (S ) are known as cells of
a n
level m. We note that, for each choice of m, S is the union of the nm cells of
n
level m.
The self–similar fractals that are considered in this work are defined next.
Definition 1. For n ∈ N define P as the quotient space S / ∼, with the
n n
equivalence relation given by
a a a ...a bc˙∼a a a ...a cb˙
0 1 2 k 0 1 2 k
for any choice of symbols a , b and c.
j
P isatrivialspacewithonlyoneelement,P ishomeomorphictoacompact
1 2
interval in R, and P is the Sierpinski Gasket.
3
2 Graph approximations of Self-Similar Sets
Inthisandthenextsections,weconsideroneoftheself–similarsetsP defined
n
above, for an arbitrary but fixed value of n≥2.
LetV bethesetofpointsinP thathavetheformk˙ withk =1,...,n−1.
0 n
We call V the boundary of P . Likewise, for m ∈ N let V be the subset of
0 n m
P of points of the form a ...a k˙. In other words, x∈V if and only if it
n 0 m−1 m
belongs to the image of V under some m-contraction.
0
Next, we define the graphs that will approximate P .
n
Definition 2. Denote by Γ the complete graph of n vertices, with V its set
0 0
of vertices. For m ∈ N, let Γ be the graph with set of vertices V and edge
m m
relation established by requiring x to be connected with y if and only if there
exists an m-contraction T such that both points x and y are in T (V ).
a a 0
3
Wecanseethatanequivalentformulationisthattwoverticesxandy share
anedge inΓ onlywhentheir firstm symbolscoincide. Itis worthnotingthat
m
even though V ⊂V ⊂V ···, the edge relation is never preserved; this follows
0 1 2
from the fact that if x(cid:54)=y are connected in Γ , then their (m+1)-th symbols
m
cannot be equal, so that they will not be connected in Γ .
m+1
For each m ∈ N let ∆ be the graph–Laplacian on Γ . We consider the
m m
Laplacianasactingonaspacewithboundary. Moreprecisely, forareal–valued
function u defined on V and x in V \V :
m m 0
(cid:88)
∆ u(x)= (u(x)−u(y)),
m
y∼x
with the sum over all vertices y that share an edge with x; the boundary values
remain unchanged. Also u is an eigenfunction of ∆ with eigenvalue λ, if
0
∆ u(x)=λu(x), ∀x∈V \V .
m m 0
WedenotebyE (·,·)theassociatedquadraticform(knownastheenergy prod-
m
uct of the graph):
(cid:88)
E (u,v)=(∆ u,v)= (u(x)−u(y))(v(x)−v(y))
m m
x∼y
for u and v real–valued functions defined on V , and the sum being taken over
m
the pairs of vertices (x,y) that are connected to each other. Also, we use the
abbreviation E(u)=E(u,u).
3 Spectral Decimation
Let m>1, and suppose u is an eigenfunction of ∆ , with eigenvalue λ .
m−1 m−1
We will show that it is always possible to extend this function to the domain
V so that it will be an eigenfunction of ∆ (with not the same eigenvalue).
m m
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 that u is an eigenfunction of ∆ with eigenvalue λ ; we aim to
m m
write the values of u in V \V in terms of its values in V . Without
m m m−1 m−1
loss of generality, we can restrict ourselves to the set V ∩T (P ) for a fixed
m a n
(m−1)-contraction T ; thisisbecausetheverticesofΓ thatbelongtotheset
a m
(V \V )∩T (P )arenotconnectedtoanyverticesoutsidethecellT (P ).
m m−1 a n a n
Denote the elements of this set by
x =abc˙. b,c=0,...n−1. (1)
b,c
It is clear that x = x , and also that x ∈ V if and only if b = c.
b,c c,b b,c m−1
This is shown in Figure 2.
4
Figure 2: A cell of level m−1 of a graph Γ , approximating P .
m 4
For each point x ∈V ∩T (P ) define the sets of vertices
r,s m a n
F ={x | i(cid:54)=j,{r,s}∩{i,j}=(cid:54) ∅}
r,s i,j
G ={x | i(cid:54)=j,{r,s}∩{i,j}=∅}
r,s i,j
Thisis,F isthesetofvertices(notinΓ )thatareconnectedtothevertex
r,s m−1
x in Γ , and G is the set of vertices (not in Γ , either) that are not
r,s m r,s m−1
connected to it.
For every r (cid:54)=s we have
(cid:88)
(2(n−1)−λ )u(x )=u(x )+u(x )+ u(x ). (2)
m r,s r,r s,s i,j
Fr,s
Addingthisupoverallthepossiblevaluesofr ands,andrearrangingterms
yields
n−1
(cid:88) (cid:88)
(2−λ ) u(x )=(n−1) u(x ),
m r,s j,j
r(cid:54)=s j=0
which for any fixed a(cid:54)=b can also be written in the form
 
n−1
(cid:88) (cid:88) (cid:88)
(2−λm)u(xa,b)+ u(xi,j)+ u(xi,j)=(n−1) u(xj,j).
Fa,b Ga,b j=0
This, together with (2) allows us to express the sum of the values in G in
a,b
termsofu(x )andthevaluesatpointsinV ; namely, providedλ (cid:54)=2, we
a,b m−1 m
5
have that
n−1
(cid:88) (n−1) (cid:88)
u(x )=u(x )+u(x )−(2n−1−λ )u(x )+ u(x ). (3)
i,j a,a b,b m a,b 2−λ j,j
m
Ga,b j=0
Next, we will take the sum of the same terms, but only over the x inside
i,j
thesetF forfixedvaluesa(cid:54)=b. Inordertodothis,wenotethatF contains
a,b a,b
two complete graphs with n−1 vertices (one with the points x and one with
i,a
thepointsx ),andthesecompletegraphsarepairwiseconnectedtoeachother
i,b
(x ∼ x ). From this, it is clear that each x ∈ F is connected to other
i,b i,a i,j a,b
n−1verticesinF . Also,eachx ∈G isconnectedtoexactlyfourvertices
a,b i,j a,b
inF . FortheverticesinV wenotethatx isconnectedton−2vertices
a,b m−1 j,j
in x ∈F if j =a,b and to only two vertices otherwise.
i,j a,b
From the preceeding discussion it follows the equality
(cid:88) (cid:88)
(n−λ ) u(x )=4 u(x )+(n−2)(u(x )+u(x ))
m i,j i,j a,a b,b
Fa,b Ga,b
(cid:88)
+2(n−2)u(x )+2 u(x ) (4)
a,b j,j
j(cid:54)=a,b
Considertheexpresiongivenby(2)for{a,b}={r,s},multiplyitbyn−λ ,
m
and substitute equality (4) into it; this gives after arranging terms
(cid:88)
(λ2 −(3n−2)λ +2(n2−2n+2))u(x )=4 u(x )
m m a,b i,j
Ga,b
(cid:88)
+2 u(x )+(2(n−1)−λ )(u(x )+u(x )).
j,j m a,a b,b
j(cid:54)=a,b
We want to get rid of the terms corresponding to G , so we replace it by (3).
a,b
After straightforward computations, we can see that for λ (cid:54)=2
m
(λ2 −(3n+2)λ +2n(n+2))u(x )=
m m a,b
(cid:0)λ2 −2(n+2)+8n(cid:1)(u(x )+u(x ))+2(2n−λ )(cid:80) u(x )
= m a,a b,b m j(cid:54)=a,b j,j .
2−λ
m
The quadratic equation for λ in the left hand side has roots n+2 and 2n.
m
The one in the right hand side has roots 4 and 2n. This gives us the following
expression for u(x ) in terms of the values of u in V :
a,b m−1
(cid:80)
(4−λ )(u(x )+u(x ))+2 u(x )
u(x )= m r,r s,s j(cid:54)=r,s j,j (5)
r,s (2−λ )((n+2)−λ )
m m
valid for any eigenvalue λ (cid:54)=2,n+2,2n.
m
6
For λ =0 this reduces to
m
2 1 (cid:88)
u(x )= (u(x )+u(x ))+ u(x ). (6)
r,s n+2 r,r s,s n+2 j,j
j(cid:54)=r,s
It is clear from the construction that if u(x ) is defined by (5) and λ is
a,b m
given by (10), then we have that
∆ u(x )=λ u(x ). (a(cid:54)=b).
m a,b m a,b
ItremainstoverifythatthisisvalidaswellinV . Ofcourse, thiscannot
m−1
betrueforarbitraryvaluesofλ ,butonlyatmostforspecificvaluesdepending
m
on λ ; we will find those values in what follows.
m−1
Take a point in V , say
m−1
x =ap˙, a=a ...a .
p,p 0 m−1
Suppose that a = q is the last symbol in a that is different from p; we can
k
assume that such symbol exists, since otherwise x would be in the boundary
p,p
V . With this, the point x can also be written in the form
0 p,p
x =a(cid:48)q˙, a(cid:48) =a ...a pq...q
p,p 0 k−1
withthenecessarynumberofq’stomakea(cid:48) awordoflengthm−1. Hence,x
p,p
is in exactly two different (m−1)-cells: T (P ) and T (P ), corresponding to
a n a(cid:48) n
each one of its two representations.
Denote by x(cid:48) the points in T (P )∩V , defined as in (1) for the points
r,s a(cid:48) n m
in T (P )∩V ; in particular x = x(cid:48) (see Figure 3). The value of u in the
a n m p,p q,q
pointsx(cid:48) isgivenbytheanalogueofequation(5). Thevertexx isconnected
r,s p,p
inΓ tothe2(n−1)pointsoftheformx andx(cid:48) ,fromwhichitfollowsthat
m p,j q,j
u is an eigenfunction of ∆ with eigenvalue λ , if and only if (5) holds for all
m m
x ∈V \V and the following equality holds for all x ∈V :
r,s m m−1 p,p m−1
(2(n−1)−λ )u(x )= (cid:88) (cid:0)u(x )+u(x(cid:48) )(cid:1). (7)
m p,p p,j q,j
i(cid:54)=p,j(cid:54)=q
On the other hand, since we know that u is an eigenfunction of ∆ with
m−1
eigenvalue λ , we also have that:
m−1
(2(n−1)−λ )u(x )= (cid:88) (cid:0)u(x )+u(x(cid:48) (cid:1). (8)
m−1 p,p i,i j,j
i(cid:54)=p,j(cid:54)=q
Replacing each term in the right hand side of (7) by its expression given by
(5) we can see that
2(n−1)(4−λ )u(x )+(2n−λ )
(2(n−1)−λ )u(x )= m p,p m ,
m p,p (2−λ )((n+2)−λ )
m m
7
x'00
x'01 x'02
x00=x'11 x'12 x'03 x'22
x01 x'13 x'23
x03
x'33
x11 x03 x02 x33
x12 x23
x22
Figure 3: Two cells of level m−1 intersecting in a vertex of Γ .
m
and using (8) this gives
[2(n−1)(4−λ )+(2n−λ )(2(n−1)−λ )]u(x )
u(x )= m m m p,p
p,p (2(n−1)−λ )(2−λ )((n+2)−λ )
m m m
Taking u(x )(cid:54)=0, and cancelling out, after computations the above equal-
p,p
ity reduces to the quadratic
λ2 −(n+2)λ +λ =0, (9)
m m m−1
which in turn gives the following recursive characterization of the eigenvalues:
(cid:112)
(n+2)± (n+2)2−4λ
m−1
λ = (10)
m 2
Since this procedure can be reversed, we have proved the following result.
Theorem 1. Let λ (cid:54)= 2,n+2,2n, and let λ be given by (9). Suppose
m m−1
u is an eigenfunction of ∆ with eigenvalue λ . Extend u to V by (5).
m−1 m−1 m
Then u is an eigenfunction of ∆ with eigenvalue λ . Conversely, if u is an
m m
eigenfunction of ∆ with eigenvalue λ (cid:54)= 2,n+2,2n, then its restriction to
m m
V must be an eigenfunction of V with eigenvalue λ for either choice
m−1 m−1 m−1
in (10).
8
4 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 called renormalization problem
for the fractal (e.g. [10], Chapter 4); roughly, this consists in normalizing the
graph energies in Γ in order to obtain a self–similar energy in the fractal by
m
taking the limit. This can be achieved if the energies are such that they remain
constant for each harmonic extension from Γ to Γ . Below, we do this for
m m+1
the P sets.
n
Definition 3. For a given function u with domain V , we call the extension
m−1
of u to V given by (6) its harmonic extension.
m
The next result, gives the explicit solution of the renormalization problem
the for P .
n
Theorem 2. Let u:V →R arbitrary, and let u(cid:48) :V →R be its harmonic
m−1 m
extension. Then
n
E (u(cid:48))= E (u)
m n+2 m−1
Proof. Note that the energy at level k of a given function equals the sum
of the energies at all the k(cid:48)–cells for any k(cid:48) ≤ k, since different cells share
no edges. This allows to restrict ourselves to one fixed m−1–cell both while
considering E (u(cid:48)) and E (u). We use the notation of the previous section
m m−1
for the vertices of Γ in this cell, and write E˜ for the energy restricted to this
m
cell. We can readily see that
E˜ (u)=(cid:88)(u(x )−u(x ))2.
m−1 i,i j,j
i(cid:54)=j
n−1
(cid:88) (cid:88)
=(n−1) u2(x )−2 u(x u(x ,j)). (11)
i,i i,i i
i=0 i(cid:54)=j
In order to evaluate the energy E˜ we consider first the edges joining vertices
m
in V with vertices in V \V : The edge joining the vertex x with the
m−1 m m−1 a,a
vertex x contributes to the energy by
a,k
 2
1 (cid:88)
(u(xa,a)−u(xa,k))2 = (n+2)2 nu(xa,a)−2u(xk,k)− u(xj,j) .
j(cid:54)=a,k
When adding up this over all possible pairs a (cid:54)= k, each x will appear n−1
r,r
timesasthex ,anothern−1timesasthex and(n−1)(n−2)asoneofthe
a,a k,k
x ’s. Each pair double product 2x x will appear twice for {r,s}={a,k},
j,j r,r s,s
2(n−2)timesfor{r,s}={a,j}forsomej,also2(n−2)timesfor{r,s}={k,j}
forsomej,andfinally(n−2)(n−3)timeswhenbothr andsareoneofthej’s.
9
All this implies that the contribution to the energy from these edges is, after
simplification:
(cid:88)
(u(x )−u(x ))2
a,a a,k
a(cid:54)=k
(n−1)(n2+n+2)n(cid:88)−1 2(n2+n+2)(cid:88)
= u2(x )− x x .
(n+2)2 i,i (n+2)2 i,i j,j
i=0 i(cid:54)=j
(12)
On the other hand, the contribution from the edge that joins the vertices x
a,b
and x (in V \V ) equals
a,c m m−1
1
(u(x )−u(x ))2 = (u(x )−u(x ))2.
a,b a,c (n+2)2 b,b c,c
Taking the sum over all of the vertices in V \V yields
m m−1
(cid:88)
(u(x )−u(x ))2
a,b a,c
a(cid:54)=b(cid:54)=c
n−1
(n−1)(n−2) (cid:88) 2(n−2)(cid:88)
= u2(x )− u(x )u(x ). (13)
(n+2)2 i,i (n+2)2 i,i j,j
i=0 i(cid:54)=j
From (12) and (13) it follows that the total energy of the cell is
n−1
E˜ (u(cid:48))= n(n−1) (cid:88)u2(x )−2n(n+2). (14)
m n+2 i,i
i=0
This, together with (11) gives
n
E˜ (u(cid:48))= E˜ (u).
m n+2 m−1
Taking this result for all the m−1-cells concludes the proof.
(cid:3)
Definition 4. The energy in P is given by
n
(cid:18)n+2(cid:19)m
E(u)= lim E (u).
m→∞ n m
The domain of E(·) being the space D(n) of functions such that the energy is
finite. Write D(n) for the subspace of D(n) of functions that vanish on the
0
boundary. The energy product E(u,v) can be recovered by the polarization iden-
tity.
Let µ be a self–similar measure in P , the Laplacian ∆ is given by:
n µ
10