The spectral decimation of the Laplacian on the Sierpinski gasket

The spectral decimation of the Laplacian on theSierpinski gasket

Nishu Lal

University of California, Riverside

Fullerton College

February 22, 2011

1 Construction of the Laplacian ∆ on the Sierpinski gasketFind a sequence of graphs Γm

Define the graph Laplacian ∆m on each Γm

The Laplacian on SG is ∆ = limm→∞ cm∆m

2 The spectrum of the Laplacian on SGOutline for decimation methodThe eigenvalue diagram

3 The spectral zeta function

Nishu Lal (UCR) The spectral decimation of the Laplacian on the Sierpinski gasketFebruary 22, 2011 2 / 27

What is the Sierpinski gasket, SG?


Interesting facts about SG

The dimension of the gasket is log 3log 2 ≈ 1.5849.

The area of SG is 0.

The boundary is of infinite length (the circumference is infinity).

It is connected.


The construction of the Sierpinski gasketWe study calculus on fractals by the approach of using discreteapproximations. Consider three contraction mappings for SG:

Fi (x) =1

2(x − qi ) + qi

for i = 0, 1, 2 where {q0, q1, q2} are the vertices of a triangle.


Let S(R2) be the space of nonempty compact subsets of R2.S(R2) is a complete metric space with Hausdorff metric.

Hausdorff metric

The Hausdorff metric is a distance function on S(R2):

dH(A,B) = inf{ε > 0 : B ⊆ Aε,A ⊆ Bε}

where Aε = {x ∈ R2 : d(x ,A) ≤ ε}.


Define F : S(R2) → S(R2) by F (A) = F0(A) ∪ F1(A) ∪ F2(A). F is acontraction mapping on S(R2).

Definition

The Sierpinski gasket, SG ∈ S(R2) is a unique fixed point of F or

SG =2⋃

i=0

Fi (SG ) = F (SG )

that is, SG is a self-similar set.


The graph approximation Γm → SGThe Sierpinski gasket is approximated by a sequence of graphs.

Definition

Let Γ0 be the graph on V0. Then we define

Γm :=2⋃

i=0

Fi (Γm−1).


Let Γ0 be the graph on V0 where V0 = {q0, q1, q2}. Then the verticesof Γm are obtained inductively by

Vm :=2⋃

i=0

Fi (Vm−1).

Define V∗=⋃

m>0 Vm.

V∗ is dense in SG.

So it is enough to use the sequence of finite sets Vm to approximateSG.


The graph Laplacian on Γm

Definition

Points x and y are called neighbors in Γm if there is an m-cell containingboth or x∼

my .

Definition

Let u : Vm → R be a function. We define the graph Laplacian on Γm by

∆mu(x) :=∑x∼my

(u(x)− u(y))

where x ∈ Vm�V0.


Goal: define ∆ on SG from ∆m on Γm

We define the pointwise formulation for the Laplacian ∆ on SG as therenormalized limit of ∆m

∆u(x) := limm→∞

cm∆mu(x)

Now the goal is to find the renormalization factor cm.


We use the concept of graph energy to define the Laplacian

Graph Energy on Γm → SG

Let u : Vm→R be a function. The graph energy of u is

Em(u) = r−m∑x∼my

(u(x)− u(y))2

where r = 35 .

For any continuous function u on SG, Em(u|Vm) is a monotone increasingsequence of m.

Graph energy on SG

E(u) = limm→∞

r−m∑x∼my

(u(x)− u(y))2


To explore the relationship between Em(u) and Em+1(u), we consider thefollowing:

Given u := Vm → R, extend u to Vm+1.

There are many extensions, but there is one unique extension u thatminimizes Em+1(u).

u is called the harmonic extension.

The harmonic extension satisfies 15 -25 rule at the new points

Vm+1�Vm.

Then we get Em+1(u) = Em(u) for ∀ u.


Formulation of the Laplacian on SG

To define the Laplacian from the graph energy requires the choice of aself-similar measure µ.

Weak formulation of the Laplacian ∆µ

Let u ∈ domE and f ∈ C (SG ). Then u ∈ dom∆µ and ∆µu = f if

E(u, v) = −∫SG

fvdµ

∀ v ∈ domE vanishing on the boundary points.


finally, the Laplaican...

We can derive the pointwise formulation of the Laplacian from the weakformulation.

The Laplacian on SG is

∆µu(x) =3

2lim

m→∞5m∆mu(x).


The normalized discrete Laplacian

From now on, we will discuss the normalized discrete Laplacian. It isdefined as follows: for u : Vm → R with u vanishing on V0, the normalizedLaplacian on Γm is

∆mu(x) :=1

4

∑x∼my

(u(x)− u(y))

where x ∈ Vm�V0.


The spectrum of the Laplacian on SG

Obtain solutions of the eigenvalue equation

−∆u = λu

on SG as limits of solutions of the discrete version

−∆mum = λmum

on Vm�V0.

Construct λm+1 from λm via a polynomial R(z).


Outline of the decimation method

If u is an eigenfunction of −∆m+1 with eigenvalue λ, that is,−∆m+1u = λu and λ /∈ B, then

−∆m(u|Vm) = R(λ)u|Vm

where B = {54 ,12 ,

32} is the set of forbidden eigenvalues.

If −∆mu = R(λ)u and λ /∈ B then there exists a unique extension uof u such that

−∆m+1(u) = λ(u).


The special polynomial R(z)

The decimation method relates the eigenvalues of successive graphLaplacians by the polynomial R(z) = z(5− 4z).

Given the eigenvalues of −∆0 to be {0, 32}. To obtain eigenvalues of−∆1, we consider the inverse images of R(z):

R−(z) = 5−√25−16z8 .

R+(z) = 5+√25−16z8 .


The eigenvalue diagram

−∆ 0

−∆1

−∆

−∆ 2

3

−∆4

Operators Eigenvalues

5

4

5

4

5

4

5

4

[1]

0

[1]

0

[1]

0

[1]0

0

2[2]

3

3

4[2]

+[2] [2]

[2]

−

[2]

2

3[3]

3

4[3]

1

2

R(3/4)+

[3]

R(3/4)

[3]−

3

2[6]

3

4[6]

2

1

R(3/4)+[6]

R(3/4)−[6]

4

5[1]

+R(5/4)

[1] [1]

R(5/4)−

3

2[15]

3

[15]

4

1

2

1

2

5

4[4]

R(5/4) R(5/4)+ −

[4] [4]

R R(3/4)+ +

R R(3/4)−+

[1]

R (3/4) R (3/4)


Eigenvalues of the Laplacian ∆µ

Every eigenvalue of ∆µ has a form

5i limm→∞

5m(R−)m−jRω(z0)

where ω is a word with |ω| = j taking values in {−,+} and (R−)m−j is the(m-j)th iteration power of R− and z0 = 3

4 ,54 .

If R−(z) < z then Rn−(z) is decreasing so it converges to a fixed point

of R− which is 0.

The limit exists if R− is applied after finite steps.


The spectral zeta function

Definition

Suppose A is a non-negative self-adjoint operator with discrete spectrumand positive eigenvalues {λj}, then its spectral zeta function is

ζA(s) =∑λj 6=0

(λj)−s/2.

Example

Consider the Dirichlet Laplacian on [0,1], A = −d2

dx2.

Eigenvalues: λ = π2j2, j = 1,2,.....The spectral zeta function is

ζA(s) =∑∞

j=1(π2j2)−s/2

= π−s∑∞

j=1 j−s

= π−s/2ζ(s).


The spectral zeta funtion associated to SG

Alexander Teplyaev discovered the product structure of the spectralzeta function of the Laplacian on SG that involves a geometric partand a new zeta function associated with the polynomial R(z). Thiszeta function has many interesting properties...

A similar product structure of the spectral zeta function was studiedearlier by Michel Lapidus for fractal strings and can now be viewed asa special case of this situation.


Fractal String

Example

A fractal string L is a disjoint collection of open intervals of length `k .Define its geometric zeta function to be

ζL(s) =∞∑k=1

(λk)s .

Let L =⋃

Ik with |Ik | = `k and let A = −d2

dx2be the Dirichlet Laplacian on

L. σ(A) =⋃∞

k=1 σ(A; Ik) = {π2n2

`2k: n ≥ 1, k ≥ 1}.

The spectral zeta function of A:

ζA =∑

k,n≥1(π2n2

`2k)−s/2 = π−s

∑∞k=1 `

sk

∑∞n=1 n−s .

Theorem

Lapidus: ζA(s) = π−sζL(s)ζ(s), where ζ(s) is the Riemann zeta function.


Current research

In order to generalize the decimation method for a certain class offractals, Sabot introduced a function of several complex variablescalled the renormalization map.

It is interesting to study the spectral zeta function of other self-similarsets in similar setting, in particular, its product structure in terms ofthe zeta function associated with the renormalization map.

Thus far, jointly with Michel Lapidus, we have looked at theSturm-Liouville operator of the form d

dµddx on the half real line

(blow-up of a unit interval). We established the spectal zeta functionin terms of a renomalization map induced from the decimationmethod discovered by Sabot. The dynamics of an invariant curve ofthe renormalization map is used to compute the spectrum of theoperator. We define the zeta function ζf associated with therenormalization map f and relate it to the spectral zeta function ζspof the Sturm-Liouville operator via a product formula with ahyperfunction.


Fukushima M, Shima T., On the spectral analysis for the Sierpinskigasket Potential Analysis 1, (1992) 1-35.

Kigami J., Harmonic calulus on p.c.f self-similar sets. Trans. Am.Math. Soc. 335, (1993) 721-755.

Kigami J.,Lapidus M.L., Weyl’s problem for the spectral distributionof Laplacian on p.c.f self-similar fractals Comm. Math. Phys. 158(1),(1993) 93-125.

Rammal R., Spectrum of harmonic exciations on fractals J. dePhysique 45, (1984) 191-206.

Sabot C., Integrated density of states of self-similar Sturm-Liouvilleoperators and holomorphic dynamics in higher dimension, Ann. I. H.Poincare, 37(3) (2001).

R. Strichartz, Differential equations on fractals: a tutorial, PincentonUniversity Press, 2006.

A. Teplyaev, Spectral zeta function of fractals and the complexdynamics of polynomials; Trans. Amer. Math. Soc., 359(2006),4339-4358.Nishu Lal (UCR) The spectral decimation of the Laplacian on the Sierpinski gasketFebruary 22, 2011 27 / 27

The spectral decimation of the Laplacian on the Sierpinski gasket

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