Eigenvalue Data on the Sierpinski Carpet

Matthew Begue

November 3, 2010

Construction of SC

The Sierpinski Carpet, SC , is constructed by eight contraction mappings.The maps contract the unit square by a factor of 1/3 and translate to oneof the eight points along the boundary. SC is the unique nonemptycompact set satisfying the self-similar identity

SC =7⋃

i=0

Fi (SC ).

F0 F1 F2

F3

F4F5F6

F7

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Constructing the Laplacian

The Laplacian ∆ on SC was independently constructed by Barlow & Bass(1989) and Kusuoka & Zhou (1992).In 2009, Barlow, Bass, Kumagai, & Teplyaev showed that both methodsconstruct the same unique Laplacian on SC .

We will be following Kusuoka & Zhou’s approach in which we consideraverage values of a function on any level m-cell.

We approximate the Laplacian on the carpet by calculating the graphLaplacian on the approximation graphs where verticies of the graph arecells of level m:

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Constructing the Laplacian

∆mu(x) =∑y∼xm

(u(y)− u(x)).

����

~~a

b

c

d����

~~x

x

x ′ y

z

For example the graph Laplacian of interior cell a is∆mu(a) = −3u(a) + u(b) + u(c) + u(d).For boundary cells, we include its neighboring virtual cells.eg: ∆mu(x) = −4u(x) + u(y) + u(z) + u(x) + u(x ′).

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Construction of the Laplacian

The Laplacian on the whole carpet is the limit of the approximating graphLaplacians

∆ = limm→∞

r−m∆m

where r is the renormalization constant r = (8ρ)−1.

So far, ρ has only been determined experimentally. ρ ≈ 1.251 andtherefore 1/r ≈ 10.011.

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Harmonic Functions

A harmonic function, h, minimizes the graph energy given a functiondefined along the boundary as well as satisfying ∆h(x) = 0 for all interiorcells x .The boundary of SC is defined to be the unit square containing all of SC .Example: Set three edges of the boundary of SC to 0 and assign sinπxalong the remaining edge and extend harmonically.

sinπx

0 0

0Matthew Begue () Eigenvalue Data on the Sierpinski Carpet November 3, 2010 6 / 25

More Harmonic Functionssin 2πx

0 0

0sin 3πx

0 0

0Matthew Begue () Eigenvalue Data on the Sierpinski Carpet November 3, 2010 7 / 25

Boundary Value Problems

We also wish to solve the eigenvalue problem on the Sierpinski Carpet:

−∆u = λu

We have two types of boundary value problems:

Neumann∂nu |∂SC = 0

Corresponds to even reflectionsabout boundary.

ie: x = x

Dirichletu|∂SC = 0

Corresponds to odd reflections aboutthe boundary.

ie: x = −x

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∆mu(x) =∑y∼xm

(u(y)− u(x))

����

~~a

b

c

d����

~~x

x

x ′ y

z

Therefore the Laplacian operator is determined by 8m linear equations.

This can be represented in an 8m square matrix.

The matrix is created in MATLAB and the eigenvalues andeigenfunctions are calculated using the built-in eigs function.

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Some eigenfunctions

Neumann: ∂nu|∂SC = 0 Dirichlet: u|∂SC = 0

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Refinement

On level m + 1 we expect to see all 8m eigenfunctions from level mbut refined.The eigenvalue is renormalized by r = 10.011.

Figure: φ(4)5 and φ

(5)5 with respective eigenvalues λ

(4)5 = 0.00328 and

λ(4)5 = 0.000328.

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Miniaturization

Any level m eigenfunction and eigenvalue miniaturizes on the level m + 1carpet. It will consist of 8 copies of φ(4) or −φ(4).

Figure: φ(4)4 and φ

(5)20 with respective eigenvalues λ

(4)4 = λ

(5)20 = 0.00177.

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Describing the eigenvalue data

Eigenvalue counting function: N(t) = #{λ : λ ≤ t}

N(t) is the number of eigenvalues less than or equal to t. Describesthe spectrum of eigenvalues.

We expect the N(t) to asymptotically grow like tα as t →∞ whereα = log 8/ log 10.011 ≈ 0.9026.

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N(t) = #{λ : λ ≤ t}

Weyl Ratio: W (t) = N(t)tα α ≈ 0.9

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Neumann vs. Dirichlet eigenvalues

By the min-max property, we can say that λ(N)j ≤ λ(D)

j for each j .

Therefore, N(D)(t) ≤ N(N)(t).

What is the growth rate of N(N)(t)− N(D)(t). We suspect there issome power β such that N(N)(t)− N(D)(t) ∼ tβ.

β = log 3log 10.011 ≈ 0.4769

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N (N)(t)− N (D)(t)

Matthew Begue () Eigenvalue Data on the Sierpinski Carpet November 3, 2010 17 / 25

N(N)(t)−N(D)(t)tβ

A stronger periodicity is apparent here.

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Fractafolds

We can eliminate the boundary of SC by gluing its boundary in specificorientations. We examined three types of SC fractafolds:

@

@

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�

� �@ @ � �@ @

@ @� �

� �@ @

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Torus Klein Bottle Projective Space

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Some Eigenfunctions for the Fractafolds

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How to define the normal derivative on the boundary of SC

We wish to define ∂nu on ∂SC so that the Gauss-Green formula holds:

E(u, v) = −∫

SC(∆u)v dµ+

∫∂SC

(∂nu)v dµ′.

We know that

Em(u, v) =1

ρm

∑x∼y

(u(x)− u(y))(v(x)− v(y))

and

−∆mu(x) =8m

ρm

∑x∼y

(u(x)− u(y)).

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Sketch of how ∂nu is defined

Let x be a point on ∂SC and xm be the m-cell containing x . We can usethe equations from the previous slide to find ∂mu remembering to givespecial treatment to cells on the boundary of SC because we mustincorporate their virtual cells.After much rearrangement we obtain∫

∂SCv∂nu dx =

2 · 3m

ρm

∑xm∼∂SC

v(xm)(f (x)− u(x))1

3m

which lets us define the normal derivative as:

∂nu(x) = limm→∞

2 · 3m

ρm(u(x)− u(xm)).

The normal derivative most likely only exists as a measure.

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Rates of Convergence

Let us return to the harmonic extension of SC with sinπnx defined alongone edge of the boundary. We wish to describe how the values of the cellsalong the bottom edge of the boundary approach 0.

For every cell, xm, along the boundary we take the value of its two parentcells, xm−1 and xm−2. They decay exponentially. We find that exponentialrate for each of the 3m cells along the boundary.

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Calculated Decay Rates

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Website

More data is available (and more to come) onwww.math.cornell.edu/∼reu/sierpinski-carpetincluding:

List of eigenvalues and pictures of eigenfunctions for both Dirichletand Neumann boundary value problems.

Eigenvalue counting function data & Weyl ratios.

Eigenvalue data on fractafolds and covering spaces of SC .

Trace of the Heat Kernel data.

Dirichlet and Poisson kernel data.

All MATLAB scripts used.

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