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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?
Nishu Lal (UCR) The spectral decimation of the Laplacian on the Sierpinski gasketFebruary 22, 2011 3 / 27
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.
Nishu Lal (UCR) The spectral decimation of the Laplacian on the Sierpinski gasketFebruary 22, 2011 4 / 27
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.
Nishu Lal (UCR) The spectral decimation of the Laplacian on the Sierpinski gasketFebruary 22, 2011 5 / 27
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}.
Nishu Lal (UCR) The spectral decimation of the Laplacian on the Sierpinski gasketFebruary 22, 2011 6 / 27
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.
Nishu Lal (UCR) The spectral decimation of the Laplacian on the Sierpinski gasketFebruary 22, 2011 7 / 27
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).
Nishu Lal (UCR) The spectral decimation of the Laplacian on the Sierpinski gasketFebruary 22, 2011 8 / 27
Nishu Lal (UCR) The spectral decimation of the Laplacian on the Sierpinski gasketFebruary 22, 2011 9 / 27
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.
Nishu Lal (UCR) The spectral decimation of the Laplacian on the Sierpinski gasketFebruary 22, 2011 10 / 27
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.
Nishu Lal (UCR) The spectral decimation of the Laplacian on the Sierpinski gasketFebruary 22, 2011 11 / 27
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.
Nishu Lal (UCR) The spectral decimation of the Laplacian on the Sierpinski gasketFebruary 22, 2011 12 / 27
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
Nishu Lal (UCR) The spectral decimation of the Laplacian on the Sierpinski gasketFebruary 22, 2011 13 / 27
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.
Nishu Lal (UCR) The spectral decimation of the Laplacian on the Sierpinski gasketFebruary 22, 2011 14 / 27
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.
Nishu Lal (UCR) The spectral decimation of the Laplacian on the Sierpinski gasketFebruary 22, 2011 15 / 27
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).
Nishu Lal (UCR) The spectral decimation of the Laplacian on the Sierpinski gasketFebruary 22, 2011 16 / 27
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.
Nishu Lal (UCR) The spectral decimation of the Laplacian on the Sierpinski gasketFebruary 22, 2011 17 / 27
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).
Nishu Lal (UCR) The spectral decimation of the Laplacian on the Sierpinski gasketFebruary 22, 2011 18 / 27
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).
Nishu Lal (UCR) The spectral decimation of the Laplacian on the Sierpinski gasketFebruary 22, 2011 19 / 27
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 .
Nishu Lal (UCR) The spectral decimation of the Laplacian on the Sierpinski gasketFebruary 22, 2011 20 / 27
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)
Nishu Lal (UCR) The spectral decimation of the Laplacian on the Sierpinski gasketFebruary 22, 2011 21 / 27
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.
Nishu Lal (UCR) The spectral decimation of the Laplacian on the Sierpinski gasketFebruary 22, 2011 22 / 27
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).
Nishu Lal (UCR) The spectral decimation of the Laplacian on the Sierpinski gasketFebruary 22, 2011 23 / 27
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.
Nishu Lal (UCR) The spectral decimation of the Laplacian on the Sierpinski gasketFebruary 22, 2011 24 / 27
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.
Nishu Lal (UCR) The spectral decimation of the Laplacian on the Sierpinski gasketFebruary 22, 2011 25 / 27
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.
Nishu Lal (UCR) The spectral decimation of the Laplacian on the Sierpinski gasketFebruary 22, 2011 26 / 27
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
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