Outer Approximation of the Spectrum of a Fractal Laplacian

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arXiv:0904.3757v1 [math.AP] 24 Apr 2009

Outer Approximation of the Spectrum of a Fractal Laplacian Tyrus Berry

Steven M. Heilman

Robert S. Strichartz1

April 24, 2009 Abstract We present a new method to approximate the Neumann spectrum of a Laplacian on a fractal K in the plane as a renormalized limit of the Neumann spectra of the standard Laplacian on a sequence of domains that approximate K from the outside. The method allows a numerical approximation of eigenvalues and eigenfunctions for lower portions of the spectrum. We present experimental evidence that the method works by looking at examples where the spectrum of the fractal Laplacian is known (the unit interval and the Sierpinski Gasket (SG)). We also present a speculative description of the spectrum on the standard Sierpinski carpet (SC), where existence of a self-similar Laplacian is known, and also on nonsymmetric and random carpets and the octagasket, where existence of a self-similar Laplacian is not known. At present we have no explanation as to why the method should work. Nevertheless, we are able to prove some new results about the structure of the spectrum involving “miniaturization” of eigenfunctions that we discovered by examining the experimental results obtained using our method.

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Introduction

Laplacians arise in many different mathematical contexts; three in particular that will interest us: manifolds, graphs and fractals. There are connections relating these different types of Laplacians. Manifold Laplacians may be obtained as limits of graph Laplacians for graphs arising from triangulations of the manifold ([Colin de Verdi`ere 1998, Dodziuk and Patodi 1976]). Kigami’s approach of construction Laplacians on certain fractals, such as the Sierpinski gasket (SG), also involves taking limits of graph Laplacians for graphs that approximate the fractal ([Kigami 2001, Strichartz 1999, Strichartz 2006]). In this paper we present another connection, where we approximate the fractal from without 1 The first and second authors were supported by the National Science Foundation through the Research experiences for Undergraduates (REU) Program at Cornell University. The third author was supported in part by the National Science Foundation, Grant DMS 0652440.

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by planar domains, and attempt to capture spectral information about the fractal Laplacian from spectral information about the standard Laplacian on the domains. Thus we add an arrow to the diagram: graphsK KK r r r KK r r KK r r KK r % xrr / fractals manifolds We should point out that the probabilistic approach to constructing Laplacians on fractals also involves approximating from without, but in that case it is the stochastic process generated by the Laplacian that is approximated, so it is not clear how to obtain spectral information. We may describe our method succinctly as follows. Suppose we have a self-similar fractal K in the plane, determined by the identity [ K= Fi K (1.1) where {Fi } is a finite set of contractive similarities (called an iterated function system, IFS). Choose a bounded open set Ω whose closure contains K, and form the sequence of domains Ω0 = Ω [ Ωm = Fi Ωm−1

for m ≥ 1

(1.2)

Consider the standard Laplacian ∆ on Ωm with Neumann boundary conditions (recall that such conditions make sense even for domains with rough bound(m) ary). Let {λn } denote the eigenvalues in increasing order (repeated in case of (m) nontrivial multiplicity) with eigenfunctions {un } (L2 normalized). So (m) −∆u(m) = λ(m) n n un

(1.3)

m Of course λm 0 = 0 with u0 constant. We then hope to find a renormalization factor r such that lim rm λ(m) = λn (1.4) n m→∞

exists and lim u(m) n |K = un

m→∞

(1.5)

exists. (We have to be careful in cases of nontrivial multiplicity, and we may (m) have to adjust un by a minus sign in general). If this is the case then we may simply define a self-adjoint operator ∆ on K by −∆un = λn un

(1.6)

Of course we would also like to identify ∆ with a previously defined Laplacian, if such is possible, or at least show that ∆ is a local operator satisfying some sort of self-similarity. 2

This may seem like wishful thinking, but it is not implausible. After all, many other types of structures on fractals can be obtained as limits of structures on Ωm , so why not a Laplacian? After reading this paper, we hope the reader will agree that there is a lot of evidence that this method should work in many cases. We leave to the future the challenge of describing exactly when it works, and why. We note one great advantage of our method: it not only approximates the Laplacian, but it gives information about the spectrum. Other methods of constructing Laplacians on fractals do not yield spectral information directly. Of course, not all spectral information is immediately available. In particular, (m) asymptotic information must be lost, since we know from Weyl’s law that λn = O(n) for each fixed m, but for fractals Laplacians this is not the case. This means, in particular, that the limit (1.4) is not uniform in n. To get information about λn for large n requires taking a large value for m. In practice, our numerical calculations get stuck around m = 4. So we only see an approximation to a segment at the bottom of the spectrum. But this is already enough to reveal aspects of the spectrum that are provable. Briefly, if the fractal has a nontrivial finite group of symmetries, then every Neumann eigenfunction can be miniaturized, and so there is an eigenvalue renormalization factor R such that if λ is an eigenvalue then so is Rλ. The argument for this works for the approximating domains and also for a self-similar Laplacian on the fractal. (In fact the argument could be presented on the fractal alone, so its validity is independent of the validity of the outer approximation method, but in fact it was discovered by examining the experimental data!) So what is the evidence for the validity of the outer approximation method? First we show that it works for the case when K is the unit interval (embedded in the x-axis in the plane). In this case we can take F0 (x, y) = 21 x, 12 y and F1 (x, y) = 12 x + 12 , 12 y . If we take Ω to be the unit square, then we can compute the spectra of Ωm (rectangles) and verify everything by hand (r = 1 in this case). We do this in section 2, where we also look at different choices of Ω, producing sawtooth shaped domains, whose spectra are computed numerically. In section 3 we look at the case of SG, where the spectrum is known exactly. Here we see numerically how the spectra of the approximating domains approaches the known spectra. This computation shows that the accuracy falls off rapidly as n increases. We are also able to compare the eigenfunctions of the approximating domains with the known eigenfunctions on SG. In this case it is natural to take Ω to be a triangle containing SG in its interior since this yields connected domains Ωm . We examine how the size of the overlap influences the spectra. After the work reported in section 3 was completed, a different approach to outer approximation on SG was studied in [Blasiak et al. 2008]. In particular, different methods for choosing approximating domains are used, and a whole family of different Laplacians are studied. In section 4 we examine numerical data for some fractals for which very little had been known about the spectrum of the Laplacian, and in some cases where even the existence of a Laplacian is unknown. These examples fall outside of the

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postcritically finite (PCF) category defined in [Kigami 2001]. The first example is the standard Sierpinski carpet SC (cut out the middle square in tic-tac-toe and iterate). Here it is known that a self-similar Laplacian exists [Barlow 1995], but the construction is indirect, and uniqueness is not known. (After this work was completed, uniqueness was established in [Barlow et al. 2008].) But we also examine some nonsymmetric variants of SC for which the existence of a Laplacian is unknown. We also examine a symmetric fractal, the octagasket, where existence of a Laplacian is unknown. In all cases the spectra of the approximating regions appear to converge when appropriately renormalized. We can identify features of the spectrum, such as multiple eigenvalues, and eigenvalue renormalization factors R, and we produce rough graphs of eigenfunctions on the fractal. In particular, there is no discernible difference between the behavior in the case of the standard SC and the other examples. In section 5 we describe the miniaturization process that produces the eigenvalue renormalization factor. For this to work we need a dihedral group of symmetries of the fractal. We only deal with the examples at hand, but it is clear that it works quite generally (we also explain how it works on the square). For (m) the approximating regions, this shows how R0 λn shows up in the spectrum on Ωm+1 (the factor R0 is not the same as R). In section 6 we examine numerical data of randomly constructed variants of SC, where the existence of Laplacians is unknown . To make these carpets, we modify the construction of SC. We fix the number of squares cut out at each recursive step, but we randomly determine which squares are removed. Then, we achieve connected domains Ωm with a suitable change to the above algorithm and properly chosen parameters. Here we again see convergence of normalized eigenvalues. These random carpets are related to the Mandelbrot percolation process. See [Chayes et al. 1988] and [Broman and Camia 2008], for example. How do we compute the spectrum of the Laplacian on the approximating domain? We use a finite element method solver in Matlab, Matlab’s own pdeeig function. To do this we only need to describe the geometry of the polygonal domain Ωm . Then we either choose a triangulation (exclusive to Section 6) or let Matlab’s triangulation functions decsg and initmesh produce a triangulation and then use piecewise linear splines in the finite element method. Note that it would be preferable to use higher-order splines, at least piecewise cubic, since these increase accuracy dramatically for a fixed memory space and running time. As a concession, all of our triangulations may be further refined with the refinemesh function. The advantage of automating the triangulation is that it saves a tremendous amount of work; in particular it chooses nonregular triangulations that increase accuracy. The disadvantage is that the program usually does not pick a triangulation with the same symmetry as the domain. This means that the eigenspaces that have nontrivial multiplicity in the domain end up being split into clusters of eigenspaces with eigenvalues close but not quite equal. Since a lot of the structure of the spectrum we are trying to observe has to do with multiplicities, this forces us to make ad hoc judgements as to when we have close but unequal eigenvalues, versus multiple eigenvalues.

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Why do we deal exclusively with Neumann spectra? The main reason is that Neumann boundary conditions on the approximating domains appears to lead to Neumann boundary conditions for the Laplacian on the fractal in the case of the interval and SG, while at the same time Dirichlet boundary conditions on the approximating domains do not lead to Dirichlet boundary conditions for the Laplacian on the fractal. For example, in the case of the interval you would need to use a mix of Dirichlet and Neumann boundary conditions on different portions of the boundary. It is not at all clear what to do for other fractals. Indeed for SC it is not even clear what to choose for the boundary. The advantage of Neumann boundary conditions is that one can dispense with all notions of boundary, and define eigenfunctions simply as stationary points of the Rayleigh quotient with no boundary restrictions. All our programs, as well as further numerical data may be found on the websites www.math.cornell.edu/~thb9d [and www.math.cornell.edu/~smh82]. Finally, we note that [Kuchment and Zeng 2001] have studied similar outer approximations in the context of quantum graphs.

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2

The Unit Interval

For the unit interval I with the second derivative as Laplacian, the Neumann eigenfunctions are cos nπx with eigenvalues (πn)2 . If we take Ω to be the unit square, then Ωm is the rectangle [0, 1] × [0, 2−m ], with Neumann eigenfunctions cos nπx cos 2m kπy and eigenvalues (πn)2 + (π2m k)2 . If we restrict attention to a fixed bottom segment of the spectrum, we will only see eigenvalues with k = 0 for m large enough (specifically, eigenvalues up to L provided L ≤ (π2m )2 ). So (m) λn = λn exactly for large enough m. Of course the corresponding eigenfunctions restricted to the interval give the exact eigenfunctions of the Laplacian on the interval. Note that for each m there are many other eigenfunctions on Ωm (those with k 6= 0), but they are “blown away” in the limit. A similar analysis holds if we start with Ω equal to any rectangle with sides parallel to the axes. Note that we do not have to renormalize the spectrum, or equivalently, we can take r = 1 in (1.4). We also note how other structures on I may be approximated from corresponding structures on Ωm . For example, Lebesgue measure on I is the limit of Lebesgue measure on Ωm suitably renormalized in the sense that Z 1 ZZ m u(x, y)dxdy = lim 2 u(x, 0)dx (2.1) m→∞

Ωm

0

if u(x, y) is continuous on Ω (the result is independent of the continuous extension u(x, y) to Ω of u(x, 0) on I). A similar result holds for energy, provided we use the minimum energy extension. In other words, given f ∈ H 1 (I), let u be the minimum energy function with u(x, 0) = f (x). Then ZZ Z 1 2 2 lim 2m |∇u(x, y)| dxdy = |f 0 (x)| dx (2.2) m→∞

Ωm

0

In order to see this we expand f in a Fourier cosine series f (x) =

∞ X

ak cos πnx

(2.3)

k=0

for which we have Z

∞

1

2

|f 0 (x)| dx =

0

1X 2 (πk)2 |ak | 2

(2.4)

k=1

The minimum energy extension to Ωm is easily seen to be u(x, y) = a0 +

∞ X

ak cos πkx

k=1

with Z

2

|∇u(x, y)| dxdy = Ωm

∞ X k=1

6

cosh 2πk(2−m − y) cosh πk2−m

2

|ak | πk

sinh 2πk2−m 4 cosh2 πk2−m

(2.5)

(2.6)

Then (2.2) follows from (2.4) and (2.6). Note that we obtain the same result if we use the simpler extension u(x, y) = f (x), although this extension does not minimize energy. (The energy minimizing extension must be harmonic on the interior and satisfy Neumann boundary conditions on the portion of the boundary of Ωm disjoint from I, and this explains (2.5)). We also have a bilinear version: let Z 1

f 0 (x)g 0 (x)dx

(2.7)

(∇u · ∇v)dxdy

(2.8)

EI (f, g) = 0

and

Z Em (u, v) = Ωm

If um and vm denote the minimum energy extensions of f and g to Ωm , then lim 2m Em (um , vm ) = EI (f, g)

m→∞

We can use this to “define” a Laplacian on I via the weak formulation Z 1 EI (f, g) = − f 00 (x)g(x)dx

(2.9)

(2.10)

0

if g vanishes at 0 and 1. By the usual Gauss-Green formula Z Em (um , vm ) = (∂n um )vm ,

(2.11)

∂Ωm ∂ um , so and ∂n um = 0 on all of ∂Ωm except I, where ∂n um = − ∂y 1

Z

Em (um , vm ) = − 0

∂um ∂y

gdx.

(2.12)

Combining (2.9), (2.10) and (2.12) yields at least formally f 00 (x) = lim 2m m→∞

∂um (x, 0) ∂y

(2.13)

We can verify this by differentiating (2.5) directly (assuming f is smooth enough) to obtain ∞ X πk sinh 2πk2−m ∂um 2 (x, 0) = − (πk) ak cos πkx (2.14) ∂y cosh πk2−m k=1

and taking the limit to obtain ∞

lim 2m

m→∞

X ∂um (x, 0) = − (πk)2 ak cos πkx, ∂y

(2.15)

k=1

and this is the same value for f 00 (x) that we obtain by differentiating (2.3) directly. 7

For a less trivial example we need only to take a geometrically more interesting Ω. In particular, let Ω be a triangle with vertices (−, 0), (1 + , 0) and 1 2 , h for some choice of positive parameters and h. Then Ωm is a sawtooth region with 2m teeth, maximum height 2−m h and overlaps of length 2−m . It is not feasible to compute the Neumann spectrum of the Ωm exactly, so we use numerical methods. In Tables 2.1 and 2.3 we present the eigenvalues for several choices of parameters and level m = 2, 3, 4 (we also vary the number of refinements used in the FEM approximation). Actually the computations are done for a similar image of Ωm so that the base is exactly I, but this makes no d2 difference in the limit. The evidence suggests that we get c(, h) dx 2 in the limit for some constant that depends on the parameters. In Tables 2.2 and 2.4 we present the same data, but we normalize by dividing (m) (m) λn by λ1 . This enables us to compare the normalized eigenvalues with the expected values n2 . Note that with level m = 5 we see about a 1% deviation already at n = 6. In Figure 2.1 we show some graphs of eigenfunctions on Ωm , that approximate eigenfunctions on I. In Figure 2.2 we show the graph of an eigenfunction on Ω2 that does not approximate an eigenfunction on I. Indeed, this eigenfunction appears to be almost localized to one of the teeth. We will discuss this phenomenon in a forthcoming paper. Unfortunately, we do not know if we can define energy on I via (2.2) for a sawtooth region approximation. Indeed, we have no idea what the minimum energy extension looks like. Equilateral Triangles Sawtooth Region (height Level: 2 2 2 Refinement: 1 2 3 n 1 4.905 4.823 4.790 2 17.980 17.662 17.535 3 33.418 32.790 32.53 4 246.809 243.909 243.176 5 246.809 243.910 243.176 6 248.850 246.991 246.524 7 250.833 248.743 248.218 8 253.564 251.508 250.992 9 337.235 332.179 330.654 10 389.324 382.371 380.228 11 449.038 440.249 437.449 12 782.622 760.213 754.319 13 817.310 787.079 779.191 14 884.992 851.631 843.276 15 931.268 900.120 892.271 16 1022.576 984.632 974.972 17 1022.620 984.636 974.973 18 1023.447 995.768 988.646 19 1042.893 1004.686 995.410 20 1050.656 1012.066 1002.876 21 1269.022 1219.852 1206.751 22 1288.306 1235.661 1221.617 23 1304.295 1255.860 1242.652 24 1855.667 1749.382 1712.980 25 1878.164 1749.398 1712.983 26 1883.733 1766.473 1743.511 27 1883.965 1785.448 1761.326 28 1909.401 1821.564 1798.312 29 1948.299 1915.997

determined by requirement that triangles are equilateral, overlaps set to (2−m )/10) 2 3 3 3 3 4 4 4 4 1 2 3 4 1 2 3 4.777 17.483 32.436 242.991 242.991 246.407 248.087 250.863 330.157 379.513 436.483 752.824 777.161 841.121 890.247 972.543 972.544 986.841 993.103 1000.611 1203.162 1217.694 1238.793 1703.675 1703.676 1737.727 1755.233 1792.391 1907.606

4.868 19.097 41.513 69.950 100.984 129.818 150.737 959.592 959.592 970.250 971.305 973.587 977.148 982.108 987.564 991.919 1253.731 1315.232 1407.013 1518.109 1636.329 1747.569 1833.745

4.789 18.782 40.808 68.724 99.165 127.424 147.863 952.139 952.139 963.501 964.578 966.696 970.044 974.799 980.415 985.129 1237.690 1297.342 1386.000 1493.410 1608.281 1716.437 1798.417

4.756 18.655 40.525 68.232 98.436 126.463 146.713 950.177 950.177 961.797 962.895 964.959 968.246 972.955 978.615 983.418 1232.817 1291.832 1379.452 1485.623 1599.289 1706.278 1786.867

4.743 18.602 40.408 68.029 98.134 126.064 146.238 949.677 949.677 961.369 962.474 964.524 967.794 972.492 978.163 982.989 1231.219 1290.003 1377.254 1482.974 1596.175 1702.698 1782.782

4.828 19.218 42.900 75.398 116.012 163.836 217.674 276.002 336.966 398.337 457.640 512.105 558.576 594.214 616.898

4.749 18.903 42.191 74.140 114.066 161.053 213.915 271.161 330.967 391.162 449.268 502.535 548.015 582.929 605.004

4.717 18.776 41.906 73.635 113.283 159.935 212.408 269.220 328.563 388.285 445.913 498.708 543.791 578.409 600.254

Table 2.1: Sawtooth Unnormalized Eigenvalues, built with Equilateral Triangles There is yet another outer approximation approach to I, in which we regard 8

4 4 4.703 18.724 41.787 73.425 112.958 159.471 211.783 268.417 327.569 387.094 444.524 497.126 542.045 576.538 598.291

Figure 2.1: Sawtooth Eigenfunctions, m=2

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Equilateral Triangles Sawtooth Region (height determined by requirement Level: 2 2 2 2 3 3 Refinement: 1 2 3 4 1 2 n 1 1.000 1.000 1.000 1.000 1.000 1.000 3.665 3.662 3.661 3.660 3.923 3.922 2 3 6.813 6.798 6.793 6.790 8.527 8.522 4 50.316 50.570 50.764 50.870 14.368 14.352 5 50.316 50.570 50.764 50.870 20.742 20.709 50.732 51.209 51.463 51.585 26.665 26.610 6 7 51.136 51.572 51.816 51.936 30.962 30.878 8 51.693 52.145 52.395 52.518 197.104 198.834 9 68.750 68.871 69.025 69.118 197.104 198.834 10 79.370 79.277 79.374 79.450 199.293 201.207 91.543 91.277 91.319 91.377 199.510 201.432 11 12 159.549 157.615 157.466 157.602 199.978 201.874 13 166.621 163.185 162.658 162.697 200.710 202.573 14 180.419 176.568 176.036 176.087 201.729 203.566 15 189.853 186.622 186.264 186.372 202.849 204.739 16 208.468 204.144 203.528 203.600 203.744 205.723 17 208.477 204.144 203.528 203.600 257.521 258.466 18 208.645 206.452 206.383 206.593 270.153 270.923 19 212.610 208.301 207.795 207.904 289.006 289.437 20 214.192 209.831 209.353 209.476 311.825 311.867 21 258.709 252.912 251.913 251.880 336.108 335.856 22 262.641 256.189 255.016 254.922 358.957 358.442 23 265.900 260.377 259.407 259.339 376.658 375.562 24 378.306 362.699 357.590 356.661 25 382.892 362.702 357.590 356.661 26 384.028 366.242 363.963 363.790 27 384.075 370.176 367.682 367.455 28 389.260 377.664 375.403 375.234 29 403.940 399.970 399.354

that triangles are equilateral, overlaps set to (2−m )/10) 3 3 4 4 4 3 4 1 2 3 1.000 3.922 8.520 14.345 20.695 26.588 30.845 199.766 199.766 202.209 202.440 202.874 203.564 204.554 205.745 206.754 259.188 271.595 290.017 312.338 336.235 358.729 375.672

1.000 3.922 8.519 14.343 20.690 26.578 30.832 200.222 200.222 202.687 202.920 203.352 204.042 205.032 206.228 207.245 259.580 271.974 290.369 312.658 336.525 358.983 375.867

1.000 3.981 8.886 15.618 24.030 33.936 45.088 57.170 69.797 82.510 94.793 106.075 115.701 123.083 127.781

1.000 3.981 8.885 15.613 24.021 33.916 45.048 57.104 69.698 82.375 94.611 105.829 115.406 122.759 127.408

1.000 3.981 8.884 15.612 24.017 33.908 45.033 57.078 69.659 82.321 94.539 105.732 115.290 122.630 127.261

Table 2.2: Sawtooth Normalized Eigenvalues, built with Equilateral Triangles

Figure 2.2: Almost Localized Sawtooth Eigenfunction, m=2

10

4 4

1.000 3.981 8.884 15.611 24.016 33.905 45.027 57.068 69.644 82.299 94.509 105.693 115.243 122.576 127.201

11

Eigenvalue Data for Sawtooth Regions with different Parameters Level: 2 2 2 2 3 0.100 0.010 0.001 0.001 0.100 Height: Refinement: 2 2 2 2 2 n 1 6.256 7.155 7.550 8.017 5.093 2 23.467 26.916 29.096 31.996 20.023 3 45.256 52.154 56.017 60.827 43.675 294.835 327.674 331.873 333.511 73.960 4 319.781 356.983 368.346 377.480 107.475 5 6 388.559 435.688 465.781 504.639 139.164 7 478.443 535.783 582.431 651.190 162.497 8 696.381 851.228 1007.893 1106.376 1017.767 9 766.306 930.457 1254.199 1439.666 1027.972 10 890.920 1092.418 1677.158 1068.070 11 1003.561 1266.292 1819.137 1121.065 12 1557.709 1812.385 1175.215 13 1561.255 1931.647 1220.465 14 1608.660 1252.193 15 1616.961 1270.474 16 1618.303 1277.740 17 1865.746 1592.815 18 1996.687 1610.143 19 1641.624 20 1689.362 21 1751.504 22 1818.223 23 1871.686 24 25 26 27 28 29 30 31 32 33 7.229 28.606 63.040 108.178 159.642 210.056 248.566 1293.087 1326.104 1422.250 1568.308 1751.443 1958.367

6.996 27.616 60.656 103.746 152.707 200.594 237.152 1277.229 1304.205 1393.716 1527.962 1694.892 1882.831

7.326 29.123 64.508 111.071 163.925 214.981 253.227 1295.630 1331.648 1432.226 1587.383 1784.010

3 0.001 2 3.836 15.210 33.701 58.625 89.037 123.691 161.172 199.921 238.235 274.559 307.455 335.656 358.389 375.031 385.025 1427.143 1429.230 1437.825 1452.086 1472.750 1500.282 1535.318 1578.626 1630.678 1691.729 1761.543 1838.629 1919.939

4 0.100 2 6.782 27.041 60.510 106.725 164.994 234.348 313.478 400.647 493.580 589.334 684.176 773.516 851.985 913.808 953.571

4 0.010 2 7.060 28.154 63.022 111.207 172.017 244.478 327.260 418.579 516.082 616.709 716.553 810.781 893.700 959.146 1001.298

4 0.001 2 7.155 28.543 63.921 112.857 174.690 248.465 332.860 426.082 525.745 628.727 731.030 827.688 912.842 980.122 1023.492

4 0.001 2 2.867 11.388 25.325 44.286 67.757 95.054 125.431 158.038 192.104 226.653 260.952 294.369 326.301 356.303 384.206 409.710 432.971 453.826 472.532 489.103 503.710 516.571 527.762 537.315 545.583 552.515 558.237 562.760 566.286 568.765 570.211 1466.956 1469.178

5 0.100 2

Table 2.3: Sawtooth Unnormalized Eigenvalues

3 0.001 2

3 0.010 2 6.341 25.345 56.948 101.047 157.493 226.090 306.595 398.709 502.074 616.269 740.802 875.099 1018.500 1170.244 1329.457 1495.142 1666.162 1841.225

5 0.010 2 6.981 27.902 62.698 111.258 173.423 248.988 337.691 439.215 553.181 679.138 816.560 964.834 1123.252 1290.993 1467.118 1650.547 1840.045

5 0.001 2 3.795 15.046 33.341 58.004 88.094 122.397 159.508 197.867 235.820 271.805 304.396 332.378 354.918 371.402 381.364 1425.511 1427.493 1436.042 1450.237 1470.701 1497.967 1532.667 1575.480 1626.922 1687.201 1755.965 1831.784 1911.655 1990.465

4 0.100 3 6.768 26.987 60.387 106.506 164.649 233.848 312.794 399.751 492.443 587.934 682.499 771.560 849.767 911.372 950.989

4 0.010 3

7.011 27.957 62.567 110.372 170.667 242.467 324.432 414.783 511.178 610.581 709.130 802.054 883.758 948.192 989.668

4 0.001 3

7.061 28.160 63.035 111.227 172.045 244.511 327.290 418.596 516.074 616.660 716.448 810.605 893.449 958.826 1000.930

4 0.001 3

6.931 27.704 62.251 110.461 172.175 247.185 335.229 435.987 549.076 674.045 810.365 957.419 1114.496 1280.776 1455.315 1637.034 1824.701

5 0.001 3

12

3 0.001 2 1.000 3.975 8.805 15.160 22.375 29.343 34.564 176.844 181.760 195.488 216.666 243.504

3 0.001 2 1.000 3.957 8.721 14.965 22.084 29.059 34.386 178.882 183.449 196.750 216.955 242.290 270.915

1.000 3.965 8.786 15.283 23.211 32.245 42.015 52.117 62.105 71.574 80.149 87.501 93.427 97.765 100.371 372.037 372.581 374.821 378.539 383.926 391.103 400.236 411.526 425.095 441.010 459.210 479.305 500.502

4 0.100 2 1.000 3.987 8.922 15.737 24.328 34.554 46.222 59.075 72.778 86.896 100.881 114.054 125.624 134.740 140.603

4 0.010 2 1.000 3.988 8.927 15.752 24.366 34.630 46.356 59.291 73.102 87.356 101.499 114.846 126.591 135.862 141.833

4 0.001 2 1.000 3.989 8.933 15.773 24.414 34.725 46.520 59.548 73.477 87.870 102.167 115.676 127.577 136.980 143.041

4 0.001 2 1.000 3.973 8.834 15.448 23.636 33.158 43.754 55.128 67.011 79.063 91.027 102.684 113.823 124.288 134.022 142.918 151.032 158.307 164.832 170.613 175.708 180.194 184.098 187.430 190.314 192.732 194.728 196.306 197.536 198.401 198.905 511.714 512.490

5 0.100 2

Table 2.4: Sawtooth Normalized Eigenvalues

Eigenvalue Data for Sawtooth Regions with different Parameters Level: 2 2 2 2 3 3 0.100 0.010 0.001 0.001 0.100 0.010 Height: Refinement: 2 2 2 2 2 2 n 1 1.000 1.000 1.000 1.000 1.000 1.000 2 3.751 3.762 3.854 3.991 3.931 3.947 3 7.234 7.289 7.419 7.587 8.575 8.670 4 47.131 45.799 43.955 41.602 14.521 14.829 51.119 49.895 48.786 47.086 21.102 21.827 5 6 62.113 60.896 61.690 62.948 27.323 28.671 7 76.481 74.886 77.140 81.228 31.905 33.897 8 111.320 118.976 133.491 138.007 199.828 182.557 9 122.498 130.049 166.113 179.581 201.832 186.413 10 142.418 152.687 222.132 209.704 199.207 11 160.424 176.989 240.936 220.109 218.395 12 249.007 253.316 230.741 242.255 13 249.574 269.985 239.626 269.118 14 257.152 245.855 15 258.479 249.444 16 258.694 250.871 17 298.248 312.733 18 319.180 316.135 19 322.316 20 331.689 21 343.890 22 356.989 23 367.486 24 25 26 27 28 29 30 31 32 33 1.000 3.997 8.980 15.935 24.836 35.654 48.349 62.875 79.175 97.184 116.822 138.000 160.614 184.544 209.651 235.779 262.748 290.355

5 0.010 2 1.000 3.997 8.981 15.937 24.842 35.666 48.373 62.915 79.240 97.283 116.968 138.208 160.900 184.929 210.158 236.433 263.578

5 0.001 2 1.000 3.965 8.786 15.285 23.214 32.253 42.032 52.140 62.140 71.623 80.211 87.585 93.524 97.868 100.493 375.635 376.157 378.409 382.150 387.542 394.727 403.871 415.153 428.708 444.592 462.712 482.691 503.738 524.505

4 0.100 3 1.000 3.987 8.922 15.736 24.326 34.550 46.214 59.061 72.756 86.864 100.836 113.994 125.548 134.650 140.503

4 0.010 3

1.000 3.987 8.924 15.742 24.342 34.583 46.274 59.161 72.909 87.087 101.143 114.397 126.050 135.241 141.156

4 0.001 3

1.000 3.988 8.927 15.752 24.365 34.627 46.350 59.281 73.085 87.330 101.461 114.796 126.528 135.786 141.749

4 0.001 3

1.000 3.997 8.981 15.936 24.840 35.661 48.364 62.900 79.215 97.245 116.912 138.127 160.789 184.778 209.959 236.175 263.250

5 0.001 3

S it as the bottom line in SG. So we take Ω = SG and Ωm+1 = F1 Ωm F2 Ωm . Then Ωm is a fractafold in the sense of [Strichartz 2003] consisting of 2m cells of level m along the bottom of SG. The bottom 2m Neumann eigenfunctions of the fractal Laplacian on Ωm are obtained by the method of spectral decimation as follows. Fix a parameter j satisfying 0 ≤ j < 2m . Let xk = 2km for 0 ≤ k ≤ 2m denote the points along I where the cells of Ωm intersect, and let yk for 1 ≤ k ≤ 2m denote the top vertices of the cells (so cell number k has vertices xk−1 , xk , yk ). Then uj restricted to these points is defined by   u (x ) = 1 (cos πjx + cos πjx j k k k+1 ) 2 (2.16)  uj (yk ) = cos πjxk One can check that for a graph Laplacian ∆m on the graph {xk , yk } we have πj (2.17) −∆m uy = 2 − 2 cos m uj 2 with the appropriate Neumann conditions at the boundary points x0 , x2m . Let √ 5 − 25 − 4t φ− (t) = (2.18) 2 and (n)

Φ(t) = lim 5n φ− (t), n→∞

(2.19)

(n)

where φ− (t) denotes the n-fold composition. In particular, Φ is a smooth function with Φ(0) = 0 and Φ0 (0) = 1. Then the method of spectral decimation (See [Strichartz 2006] for a detailed explanation) says that uj may be extended to eigenfunctions of the fractal Laplacian on Ωm with eigenvalue 3 πj πj 3 (m) (n) λj = lim 5m+n φ− 2 − 2 cos m = 5m Φ 2 − 2 cos m (2.20) 2 n→∞ 2 2 2 πj 2 Now observe that 2 − 2 cos 2πj for large m so m ≈ ( 2m )

m 4 3 (m) lim λj = (πj)2 . m→∞ 5 2

(2.21)

Of course (πj)2 is the correct eigenvalue for the eigenfunction cos πjx on I, which is clearly the limit of uj as u → ∞.

13

3

The Sierpinski Gasket

Let {q0 , q1 , q2 } denote the vertices of a unit length equilateral triangle in the plane, and let Fi x = 12 (x + qi ) for i = 0, 1, 2. Then SG is the invariant set for this IFS. We take Ω to be the equilateral triangle dilated by a factor 1 + . Then Ωm is a union of 3m triangles of size 2−m that overlap in triangles of size (1 + )2−m . In Tables 3.1 and 3.2 we present the same data as in Tables 2.1 through 2.4 for this example. The multiplicities and normalized eigenvalues agree with the known values for the Neumann spectrum of the standard Laplacian on SG [Strichartz 2003]. For example, the first six distinct normalized eigenvalues on SG are 1, 5, 8.103, 10.305, 25, 31.784. So the numerical accuracy improves as we decrease but the error remains significant. (Much better accuracy is achieved in [Blasiak et al. 2008]). Nevertheless, the qualitative features of the spectrum, including high multiplicities and large gaps, are already apparent. In Figure 3.1 we show some graphs of eigenfunctions. Actual graphs of Dirichlet eigenfunctions on SG may be found in [Dalrymple et al. 1999]. In this case we know the eigenfunction renormalization factor R = 5, so we expect r = 1.25 in (1.4). The data is not inconsistent with this expectation, but it is impossible to deduce these values from the data alone. We also look at the case = 0, where the 3m triangles in Ωm intersect at single points. Thus the interior of Ωm consists of 3m disjoint triangles, and if we interpret the Neumann Laplacian on Ωm in the usual way, the spectrum would just be 3m copies of the spectrum of Ω. This is nothing like the spectrum of SG, and also it is not what we get when we use the FEM. The reason is that the spline space chosen consists of continuous functions, and this effectively couples the disjoint triangles at their junction points. Effectively this means that we are not looking at the entire Sobolev space H 1 (Ωm ), but only the subspace H01 (Ωm ) defined to be the closure of continuous functions in H 1 (Ωm ) in the Sobolev norm. In fact, functions in H01 (Ωm ) do not have to be continuous (or even bounded), since H 1 does not embed in continuous functions on R2 . They do have to satisfy some integral continuity condition (see [Strichartz 1967] for analogous results for H 1/2 on a half-line). The conclusion is that the Neumann eigenvalues (and eigenfunctions) that the FEM approximates are the stationary values (and associated functions) for the Rayleigh quotient R 2 |∇u| dx (3.1) R(u) = RΩm 2 |u| dx Ωm for some u ∈ H01 (Ωm ). Of course some of these eigenfunctions restrict to Neumann eigenfunctions on each triangle in Ωm and are continuous functions at the junction points, but it is easy to see that there are not enough of these (in fact the smallest such eigenvalue must be on the order of magnitude 4m ). We claim that all the other eigenfunctions have poles at some junction points. Indeed, consider the restriction of an eigenfunction to a triangle. Because it is a Neumann eigenfunction, it must have vanishing normal derivatives along the side of 14

Sierpinski Gasket Eigenvalue Data Level: 2 2 Refinement: 0 1 n 1 5.0727 4.8920 2 5.0729 4.8924 3 20.6394 19.9346 4 20.6560 19.9498 20.6657 19.9529 5 6 35.4198 34.0098 35.4331 34.0165 7 8 43.3830 41.5793 271.4544 266.9576 9 10 271.5749 266.9740 11 272.0985 267.1555 272.4539 268.5083 12 13 272.6653 268.5884 273.1340 269.0642 14 15 299.2086 293.1155 316.7469 309.9446 16 17 316.9947 310.0058 344.9089 336.4442 18 19 345.3793 337.1256 20 345.4605 337.1269 21 427.1256 414.3260 22 427.2329 414.3685 23 427.7069 414.5074 24 437.5025 423.7340 25 437.6399 423.7789 26 462.1666 446.9485 27 851.9381 817.0677 28 888.8229 848.6904 891.2254 849.3639 29 30 992.8633 939.1885 31 993.9410 941.2736 32 999.8911 941.9922 33 1052.0007 999.5010 34 1054.0103 999.9568 35 1088.8728 1032.6001 36 1144.7398 1084.5089 37 1150.1233 1084.9562 38 1154.9383 1087.7050 39 1156.8863 1089.5572 40 1160.0776 1090.2141 41 1170.7057 1092.5885 42 1291.1778 1215.5339 43 1293.3068 1217.2739 44 1293.9845 1217.7969 45 1297.6383 1219.9161 46 1303.1037 1221.8710 47 1306.6105 1222.1916 48 1340.1246 1241.9893 49 1343.7226 1242.7779 50 1358.2827 1245.7145 51 1383.4011 1296.6487 52 1384.0920 1296.8827 53 1408.7496 1302.9443 54 1931.2295 55 1932.7859 56 1936.4743 57 1942.3175 58 1943.1835 59 1951.4159 60 1986.5116

2 2

2 3

2 4

3 0

3 1

3 2

3 3

4 0

5 0

4.8223 4.8226 19.6622 19.6783 19.6796 33.4700 33.4733 40.8896 265.7778 265.7838 265.8297 267.5057 267.5274 268.1061 291.4105 307.9827 307.9991 333.9342 334.6768 334.6778 410.2927 410.3104 410.3490 419.2998 419.3077 441.9660 807.8753 837.8213 837.9969 923.5286 927.4429 927.5956 985.4672 985.5786 1017.3610 1067.2785 1067.3690 1068.1575 1073.8220 1074.2545 1075.6761 1192.9766 1196.5209 1196.5696 1199.2968 1199.8517 1200.0086 1216.9260 1217.0955 1217.7373 1273.2079 1273.3351 1274.8446 1881.3515 1881.4113 1882.9905 1897.9042 1898.2372 1905.0061 1939.1087

4.7946 4.7948 19.5531 19.5698 19.5704 33.2558 33.2574 40.6160 265.4780 265.4848 265.4951 267.2524 267.2580 267.8663 290.9176 307.3882 307.3930 333.1492 333.9079 333.9091 408.9367 408.9440 408.9552 417.7744 417.7749 440.2315 805.5351 834.9749 835.0193 919.3524 923.7731 923.8111 981.7155 981.7422 1013.3103 1062.9436 1062.9740 1063.1712 1069.7891 1069.9123 1071.8657 1186.5037 1190.5539 1190.5670 1193.7045 1193.9074 1193.9475 1210.4150 1210.4525 1210.6107 1266.5890 1266.6775 1267.0475 1867.0606 1867.1235 1867.5235 1886.8888 1886.9745 1894.3829 1926.7835

4.7832 4.7833 19.5080 19.5251 19.5254 33.1678 33.1685 40.5037 265.4017 265.4100 265.4123 267.1889 267.1903 267.8062 290.7663 307.1950 307.1966 332.8849 333.6483 333.6491 408.4465 408.4495 408.4531 417.2121 417.2131 439.5866 804.9468 834.2263 834.2374 918.2220 922.7715 922.7812 980.6827 980.6887 1012.2129 1061.8501 1061.8796 1061.9241 1068.7754 1068.8069 1070.9267 1184.6079 1188.8129 1188.8201 1192.0944 1192.2241 1192.2329 1208.6378 1208.6464 1208.6875 1264.6036 1264.6515 1264.7924 1863.3648 1863.4643 1863.5513 1884.1212 1884.1428 1891.7581 1923.6106

4.1689 4.1690 18.2283 18.2452 18.2457 32.1806 32.1839 41.3292 83.0086 83.0336 83.0497 83.2255 83.2406 83.3102 110.1633 119.9714 120.0147 130.5284 130.5647 130.5856 158.4240 158.4260 158.4307 179.2826 179.3092 184.3542 1071.5212 1071.7714 1071.8135 1071.8613 1072.3634 1072.6331 1073.2824 1073.5674 1073.9888 1076.4724 1076.7105 1078.5174 1089.0059 1089.4789 1090.4701 1174.0043 1174.1797 1174.6064 1181.4177 1197.4002 1197.4897 1244.6256 1246.8784 1247.1785 1292.5390 1292.7459 1318.1064 1366.3125 1366.8361 1367.4013 1373.4237 1373.5194 1376.6480 1602.4894

4.0255 4.0255 17.6218 17.6387 17.6389 31.0512 31.0522 39.8556 80.1965 80.2024 80.2096 80.4029 80.4093 80.4900 106.0977 115.5265 115.5398 125.6595 125.7034 125.7091 152.2677 152.2679 152.2696 171.6045 171.6125 176.4395 1057.8350 1057.8753 1058.3988 1059.0201 1059.1477 1059.1996 1059.2529 1059.3885 1059.4116 1059.4875 1059.5993 1059.6378 1075.0220 1075.1464 1076.1214 1154.0648 1154.1186 1154.2334 1162.3500 1177.4067 1177.4369 1220.8687 1223.3028 1223.3827 1266.7158 1266.7771 1290.8384 1335.7326 1335.8772 1336.0234 1344.3897 1344.4135 1347.7240 1559.1123

3.9697 3.9697 17.3890 17.4059 17.4060 30.6141 30.6144 39.2856 79.1253 79.1260 79.1293 79.3294 79.3318 79.4147 104.5471 113.8296 113.8336 123.8065 123.8531 123.8549 149.9313 149.9313 149.9319 168.6461 168.6488 173.3946 1053.1274 1053.1396 1053.2748 1055.7633 1055.7990 1055.8151 1055.8720 1055.8825 1055.9128 1056.0661 1056.0783 1056.0826 1071.3853 1071.4168 1072.4046 1148.4476 1148.4619 1148.4912 1156.8737 1171.6223 1171.6306 1213.9726 1216.4472 1216.4669 1259.0430 1259.0591 1282.6980 1326.6378 1326.6733 1326.7149 1335.6043 1335.6090 1338.9737 1545.2809

3.9473 3.9473 17.2965 17.3134 17.3135 30.4394 30.4394 39.0578 78.7016 78.7019 78.7031 78.9054 78.9064 78.9892 103.9332 113.1570 113.1582 123.0738 123.1212 123.1219 149.0087 149.0087 149.0089 167.4636 167.4646 172.1788 1051.9382 1051.9413 1051.975 1054.939 1054.949 1054.953 1054.965 1054.968 1054.975 1055.239 1055.244 1055.246 1070.461 1070.469 1071.462 1146.813 1146.817 1146.825 1155.237 1169.878 1169.880 1211.877 1214.360 1214.365 1256.642 1256.646 1280.138 1323.763 1323.778 1323.791 1332.801 1332.802 1336.176 1540.591

3.3372 3.3376 15.0372 15.0492 15.0518 26.2223 26.2245 33.7524 71.6959 71.6980 71.7027 71.9215 71.9232 72.0256 96.1550 106.2227 106.2293 118.5657 118.6246 118.6412 150.7050 150.7129 150.7148 159.7685 159.7975 166.8725 335.710 335.737 335.770 335.892 335.896 335.921 335.937 335.939 335.953 335.960 336.002 336.036 338.630 338.695 338.855 425.376 425.398 425.429 437.903 447.009 447.057 465.355 465.569 465.642 493.288 493.357 503.268 518.856 518.868 518.968 519.650 519.664 520.014 622.030

2.6327 2.6333 12.2130 12.2253 12.2256 20.8931 20.8956 26.8513 58.5692 58.5772 58.5823 58.7675 58.7823 58.8670 78.0084 86.4177 86.4276 97.4488 97.5097 97.5150 124.3511 124.3577 124.3641 127.3438 127.3581 133.4426 288.0482 288.0994 288.1694 288.1905 288.2102 288.2306 288.2510 288.2655 288.2700 288.2956 288.3556 288.3577 291.1745 291.3120 291.4274 371.3145 371.3541 371.3888 379.8021 388.8530 388.9220 410.6224 410.9257 410.9518 436.0154 436.0628 447.3366 466.2302 466.2960 466.3212 467.2563 467.3472 467.7506 572.2704

Table 3.1: SG Unnormalized Eigenvalues

15

Level: Refinement: n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Sierpinski Gasket Eigenvalue Data 2 2 2 0 1 2 1.0000 1.0000 4.0687 4.0720 4.0739 6.9824 6.9850 8.5522 53.5124 53.5361 53.6394 53.7094 53.7511 53.8435 58.9836 62.4410 62.4898 67.9926 68.0854 68.1014 84.2002 84.2213 84.3148 86.2458 86.2729 91.1079 167.9444 175.2155 175.6892 195.7253 195.9377 197.1107 207.3831 207.7793 214.6518 225.6650 226.7263 227.6754 228.0595 228.6886 230.7837 254.5326 254.9523 255.0859 255.8062 256.8836 257.5749 264.1816 264.8909 267.7611 272.7128 272.8490 277.7098

1.0000 1.0001 4.0750 4.0781 4.0787 6.9522 6.9535 8.4995 54.5705 54.5739 54.6110 54.8875 54.9039 55.0012 59.9177 63.3578 63.3703 68.7747 68.9140 68.9143 84.6951 84.7038 84.7322 86.6182 86.6274 91.3637 167.0221 173.4863 173.6240 191.9856 192.4119 192.5588 204.3145 204.4077 211.0805 221.6915 221.7830 222.3449 222.7235 222.8578 223.3431 248.4752 248.8309 248.9378 249.3710 249.7706 249.8362 253.8831 254.0443 254.6446 265.0564 265.1042 266.3433 394.7752 395.0934 395.8473 397.0418 397.2188 398.9017 406.0758

1.0000 1.0001 4.0773 4.0807 4.0809 6.9407 6.9414 8.4793 55.1142 55.1155 55.1250 55.4726 55.4771 55.5971 60.4297 63.8663 63.8697 69.2478 69.4018 69.4020 85.0823 85.0859 85.0939 86.9501 86.9517 91.6503 167.5288 173.7387 173.7751 191.5118 192.3235 192.3552 204.3560 204.3791 210.9698 221.3211 221.3399 221.5034 222.6781 222.7678 223.0626 247.3871 248.1221 248.1322 248.6978 248.8128 248.8453 252.3535 252.3887 252.5217 264.0247 264.0510 264.3641 390.1352 390.1476 390.4751 393.5677 393.6368 395.0404 402.1123

2 3

2 4

3 0

3 1

3 2

3 3

4 0

5 0

1.0000 1.0000 4.0781 4.0816 4.0818 6.9361 6.9364 8.4712 55.3701 55.3715 55.3737 55.7402 55.7413 55.8682 60.6760 64.1112 64.1122 69.4841 69.6424 69.6426 85.2909 85.2925 85.2948 87.1342 87.1343 91.8180 168.0085 174.1487 174.1580 191.7471 192.6691 192.6771 204.7540 204.7596 211.3437 221.6956 221.7019 221.7431 223.1233 223.1490 223.5565 247.4662 248.3110 248.3137 248.9681 249.0104 249.0188 252.4533 252.4612 252.4942 264.1694 264.1879 264.2651 389.4083 389.4215 389.5049 393.5439 393.5617 395.1069 401.8646

1.0000 1.0000 4.0784 4.0820 4.0820 6.9342 6.9343 8.4678 55.4858 55.4876 55.4880 55.8595 55.8598 55.9885 60.7887 64.2233 64.2236 69.5941 69.7537 69.7539 85.3913 85.3919 85.3927 87.2239 87.2241 91.9016 168.2851 174.4064 174.4087 191.9668 192.9179 192.9200 205.0251 205.0263 211.6169 221.9942 222.0004 222.0097 223.4420 223.4486 223.8918 247.6584 248.5375 248.5390 249.2236 249.2507 249.2525 252.6822 252.6840 252.6926 264.3826 264.3926 264.4221 389.5618 389.5826 389.6008 393.9012 393.9057 395.4978 402.1570

1.0000 1.0000 4.3724 4.3764 4.3766 7.7191 7.7199 9.9136 19.9112 19.9172 19.9210 19.9632 19.9668 19.9835 26.4248 28.7774 28.7878 31.3097 31.3184 31.3234 38.0010 38.0015 38.0026 43.0044 43.0107 44.2209 257.0248 257.0848 257.0949 257.1063 257.2268 257.2915 257.4472 257.5156 257.6167 258.2124 258.2695 258.7029 261.2188 261.3323 261.5700 281.6073 281.6494 281.7517 283.3855 287.2192 287.2407 298.5472 299.0875 299.1595 310.0401 310.0897 316.1729 327.7361 327.8617 327.9972 329.4418 329.4648 330.2152 384.3876

1.0000 1.0000 4.3775 4.3817 4.3818 7.7136 7.7139 9.9008 19.9221 19.9236 19.9254 19.9734 19.9750 19.9951 26.3564 28.6987 28.7020 31.2159 31.2268 31.2282 37.8258 37.8259 37.8263 42.6294 42.6314 43.8305 262.7838 262.7938 262.9239 263.0782 263.1099 263.1228 263.1360 263.1697 263.1754 263.1943 263.2221 263.2316 267.0533 267.0843 267.3264 286.6889 286.7022 286.7308 288.7471 292.4874 292.4949 303.2841 303.8888 303.9086 314.6733 314.6885 320.6657 331.8182 331.8541 331.8904 333.9688 333.9747 334.7970 387.3094

1.0000 1.0000 4.3805 4.3847 4.3847 7.7120 7.7121 9.8964 19.9324 19.9326 19.9334 19.9838 19.9844 20.0053 26.3364 28.6747 28.6757 31.1880 31.1998 31.2002 37.7691 37.7691 37.7693 42.4836 42.4842 43.6797 265.2927 265.2958 265.3299 265.9567 265.9657 265.9698 265.9841 265.9868 265.9944 266.0330 266.0361 266.0372 269.8921 269.9000 270.1488 289.3048 289.3084 289.3158 291.4274 295.1427 295.1448 305.8112 306.4345 306.4395 317.1648 317.1689 323.1237 334.1926 334.2015 334.2120 336.4513 336.4525 337.3001 389.2708

1.0000 1.0000 4.3819 4.3862 4.3862 7.7115 7.7115 9.8949 19.9381 19.9382 19.9385 19.9898 19.9900 20.0110 26.3303 28.6670 28.6673 31.1794 31.1913 31.1915 37.7496 37.7497 37.7497 42.4250 42.4253 43.6195 266.4966 266.4974 266.5060 267.2570 267.2593 267.2604 267.2634 267.2643 267.2660 267.3330 267.3342 267.3347 271.1893 271.1913 271.4429 290.5322 290.5332 290.5351 292.6663 296.3753 296.3758 307.0152 307.6444 307.6455 318.3560 318.3570 324.3086 335.3604 335.3642 335.3674 337.6500 337.6503 338.5050 390.2912

1.0000 1.0001 4.5059 4.5095 4.5103 7.8575 7.8582 10.1139 21.4837 21.4843 21.4857 21.5513 21.5518 21.5825 28.8128 31.8296 31.8316 35.5282 35.5458 35.5508 45.1587 45.1611 45.1617 47.8746 47.8833 50.0033 100.5955 100.6038 100.6136 100.6502 100.6512 100.6589 100.6636 100.6643 100.6685 100.6706 100.6830 100.6934 101.4705 101.4899 101.5379 127.4639 127.4704 127.4799 131.2177 133.9464 133.9608 139.4436 139.5079 139.5297 147.8138 147.8345 150.8042 155.4753 155.4788 155.5087 155.7132 155.7173 155.8221 186.3913

1.0000 1.0002 4.6390 4.6437 4.6437 7.9360 7.9370 10.1992 22.2469 22.2499 22.2519 22.3222 22.3278 22.3600 29.6307 32.8248 32.8286 37.0149 37.0380 37.0400 47.2335 47.2360 47.2384 48.3702 48.3756 50.6868 109.4121 109.4315 109.4581 109.4661 109.4736 109.4813 109.4891 109.4946 109.4963 109.5060 109.5288 109.5296 110.5995 110.6518 110.6956 141.0399 141.0549 141.0681 144.2638 147.7017 147.7279 155.9706 156.0858 156.0957 165.6158 165.6338 169.9161 177.0926 177.1176 177.1272 177.4824 177.5169 177.6701 217.3709

Table 3.2: SG Normalized Eigenvalues

16

Spectral Decimation Eigenvalues Actual Actual Normalized Unormalized 1.0000 1.0000 5.0000 5.0000 5.0000 8.1039 8.1039 10.3056 25.0000 25.0000 25.0000 25.0000 25.0000 25.0000 51.5278 35.1398 35.1398 40.5196 40.5196 40.5196 8.1039 8.1039 31.7847 31.7847 31.7847 31.7847 202.5980 202.5980 202.5980 202.5980 202.5980 202.5980 202.5980 202.5980 202.5980 202.5980 202.5980 202.5980 202.5980 202.5980 202.5980 158.9233 158.9233 158.9233 158.9233

27.1144 27.1144 135.5721 135.5721 135.5721 219.7332 219.7332 279.4291 677.8606 677.8606 677.8606 677.8606 677.8606 677.8606 1397.1457 952.7966 952.7966 1098.6658 1098.6658 1098.6658 219.7331 219.7331 861.8226 861.8226 861.8226 861.8226 5493.3291 5493.3291 5493.3291 5493.3291 5493.3291 5493.3291 5493.3291 5493.3291 5493.3291 5493.3291 5493.3291 5493.3291 5493.3291 5493.3291 5493.3291 4309.1129 4309.1129 4309.1129 4309.1129

Figure 3.1: Sierpinski Gasket (SG) Eigenfunctions, Level 3

17

the triangle. Choose a vertex of the triangle and reflect the eigenfunction evenly six times around. This yields an eigenfunction in a deleted neighborhood of the vertex. The removable singularities theorem yields the following dichotomy: either the function is unbounded or it satisfies the eigenvalue equation at the vertex. If it satisfies the eigenvalue equation at all three vertices of the triangle, then the restriction to the triangle is a Neumann eigenvalue, contrary to our assumption. It is not difficult to see that the singularities must be logarithmic poles. With this in mind, we look at the eigenvalue data in Tables 3.3 and 3.4. In contrast to our preceding computations, we do not see an apparent convergence of eigenvalues on a fixed Ωm when we increase the refinement of the triangulation. In particular the numerical values in Table 3.4 are even better than the data in Table 3.2. In other words, the poor approximations by the FEM to the actual eigenvalues on Ωm yield very good approximations to the relative eigenvalues on SG. We can even extract rather decent estimates for r = 1.25 from the data in Table 3.3 if we pair off corresponding refinements at (4) (3) different levels. For example, if we compute λn /λn using 3 refinements on level 3 and 4 refinements on level 4, the first six distinct eigenvalues yield ratios 1.246, 1.233, 1.223, 1.158, 1.128, 1.112. Of course the eigenfunctions on Ωm cannot approximate the eigenfunctions on SG, since the latter are bounded. Since we are already getting more information than we deserve, we might speculate that the eigenfunction approximation might be accurate in the complement of a small neighborhood of the junction points.

18

Sierpinski Gasket, No Overlap, Unnormalized Level: 1 1 1 Refinement: 2 3 4 n 1 3.650 3.113 2.713 3.721 3.164 2.752 2 3 70.334 70.221 70.193 70.356 70.227 70.195 4 5 70.362 70.228 70.195 82.579 80.476 79.025 6 7 82.850 80.663 79.163 8 96.289 91.743 88.598 9 212.039 210.923 210.645 10 235.397 230.816 227.972 236.040 231.194 228.236 11 12 283.145 281.338 280.886 283.383 281.397 280.901 13 14 283.654 281.464 280.918 303.428 296.864 293.464 15 16 304.318 297.305 293.740 17 310.855 303.820 299.961 497.970 492.957 491.705 18 19 499.255 493.281 491.786 20 500.036 493.472 491.833 21 526.723 515.905 511.065 22 528.063 516.482 511.415 23 568.769 547.592 536.643 24 642.681 634.411 632.344 644.193 634.780 632.435 25 26 644.399 634.848 632.454 27 646.983 635.458 632.604 28 677.294 661.371 654.619 29 681.117 662.685 655.217 30 864.923 847.924 843.639 31 888.687 867.510 860.707 890.096 868.076 861.020 32 33 938.355 918.889 914.016 34 941.159 919.527 914.172 35 944.371 920.288 914.359 36 990.412 956.321 943.405 37 995.980 958.075 944.225 38 1007.032 969.717 955.860 39 1161.208 1132.507 1125.335 40 1163.930 1133.179 1125.504 41 1167.498 1134.086 1125.730 42 1184.368 1147.961 1137.349 43 1186.310 1148.403 1137.598 44 1203.984 1162.140 1148.937 45 1384.681 1346.242 1336.679 46 1392.804 1348.470 1337.250 47 1398.054 1349.707 1337.549 48 1418.418 1369.638 1355.732 49 1424.922 1371.372 1356.319 50 1488.198 1415.530 1390.277 51 1540.507 1490.281 1477.951 52 1544.858 1491.604 1478.303 53 1547.571 1492.089 1478.407 54 1562.171 1495.591 1479.271 55 1628.290 1554.758 1529.753 56 1648.776 1560.742 1531.934 57 1847.262 1777.702 1760.374 58 1852.666 1779.553 1760.911 59 1862.961 1782.208 1761.547 60 1872.896 1793.161 1771.954

2 2

2 3

2 4

3 3

3 4

4 3

4 4

3.689 3.689 17.179 17.179 17.179 25.961 25.961 31.117 282.881 282.881 282.881 282.881 282.881 282.881 302.450 316.088 316.088 342.012 342.012 342.012 391.489 391.489 408.152 408.152 408.152 408.152 861.618 887.555 887.555 973.982 973.982 973.982 1028.867 1028.867 1063.734 1157.651 1157.651 1157.651 1157.651 1157.651 1157.651 1230.800 1247.495 1247.495 1267.855 1267.855 1267.855 1289.621 1289.621 1294.363 1294.363 1294.363 1294.363

3.103 3.103 14.285 14.285 14.285 21.402 21.402 25.509 281.270 281.270 281.270 281.270 281.270 281.270 297.037 308.066 308.066 329.003 329.003 329.003 368.308 368.308 381.181 381.181 381.181 381.181 847.027 867.807 867.807 938.678 938.678 938.678 984.272 984.272 1013.093 1131.525 1131.525 1131.525 1131.525 1131.525 1131.525 1174.160 1189.429 1189.429 1209.137 1209.137 1209.137 1231.534 1231.534 1236.613 1236.613 1236.613 1236.613 1991.484 1991.484 1991.484 1991.484 1991.484 1991.484

2.677 2.677 12.220 12.220 12.220 18.196 18.196 21.604 280.869 280.869 280.869 280.869 280.869 280.869 294.131 303.425 303.425 321.049 321.049 321.049 353.783 353.783 364.323 364.323 364.323 364.323 843.410 860.994 860.994 921.579 921.579 921.579 960.390 960.390 984.352 1125.082 1125.082 1125.082 1125.082 1125.082 1125.082 1154.340 1167.359 1167.359 1185.248 1185.248 1185.248 1206.796 1206.796 1211.852 1211.852 1211.852 1211.852 1971.705 1971.705 1971.705 1971.705 1971.705 1971.705 1999.469

2.773 2.773 13.684 13.684 13.684 21.944 21.944 27.689 63.390 63.390 63.390 63.390 63.390 63.390 78.165 85.035 85.035 95.415 95.415 95.415 110.130 110.130 114.069 114.069 114.069 114.069 1127.762 1127.762 1127.762 1127.762 1127.762 1127.762 1127.762 1127.762 1127.762 1127.762 1127.762 1127.762 1127.762 1127.762 1127.762 1198.824 1198.824 1198.824 1198.824 1209.019 1209.019 1248.438 1248.438 1248.438 1277.622 1277.622 1297.625 1342.696 1342.696 1342.696 1342.696 1342.696 1342.696 1465.787

2.364 2.364 11.644 11.644 11.644 18.643 18.643 23.498 53.398 53.398 53.398 53.398 53.398 53.398 65.625 71.278 71.278 79.778 79.778 79.778 91.738 91.738 94.922 94.922 94.922 94.922 1124.145 1124.145 1124.145 1124.145 1124.145 1124.145 1124.145 1124.145 1124.145 1124.145 1124.145 1124.145 1124.145 1124.145 1124.145 1182.609 1182.609 1182.609 1182.609 1191.017 1191.017 1223.543 1223.543 1223.543 1247.622 1247.622 1264.115 1301.218 1301.218 1301.218 1301.218 1301.218 1301.218 1401.719

2.687 2.687 13.409 13.409 13.409 21.699 21.699 27.564 66.354 66.354 66.354 66.354 66.354 66.354 84.053 92.754 92.754 106.636 106.636 106.636 128.376 128.376 134.759 134.759 134.759 134.759 311.734 311.734 311.734 311.734 311.734 311.734 311.734 311.734 311.734 311.734 311.734 311.734 311.734 311.734 311.734 386.255 386.255 386.255 386.255 394.037 394.037 421.196 421.196 421.196 439.088 439.088 450.532 474.374 474.374 474.374 474.374 474.374 474.374 529.475

2.224 2.224 11.091 11.091 11.091 17.941 17.941 22.783 54.735 54.735 54.735 54.735 54.735 54.735 69.271 76.407 76.407 87.776 87.776 87.776 105.547 105.547 110.756 110.756 110.756 110.756 253.559 253.559 253.559 253.559 253.559 253.559 253.559 253.559 253.559 253.559 253.559 253.559 253.559 253.559 253.559 312.662 312.662 312.662 312.662 318.795 318.795 340.141 340.141 340.141 354.151 354.151 363.091 381.660 381.660 381.660 381.660 381.660 381.660 424.272

Table 3.3: Sierpinski Gasket, No Overlap, Unnormalized

19

Sierpinski Gasket, No Overlap, Normalized Level: 1 1 1 Refinement: 2 3 4 n 1 1.000 1.000 1.000 2 1.019 1.016 1.014 19.268 22.557 25.875 3 4 19.274 22.559 25.876 19.275 22.559 25.876 5 6 22.622 25.851 29.131 7 22.697 25.911 29.182 26.378 29.470 32.660 8 9 58.087 67.754 77.650 64.486 74.144 84.038 10 11 64.662 74.266 84.135 77.566 90.373 103.543 12 13 77.632 90.392 103.549 77.706 90.414 103.555 14 15 83.123 95.361 108.180 83.367 95.502 108.282 16 17 85.158 97.595 110.575 18 136.417 158.351 181.258 19 136.769 158.455 181.287 20 136.983 158.517 181.305 21 144.294 165.723 188.394 22 144.661 165.908 188.523 23 155.812 175.901 197.823 24 176.060 203.790 233.102 25 176.474 203.908 233.135 26 176.530 203.930 233.142 27 177.238 204.126 233.197 28 185.542 212.450 241.313 29 186.589 212.872 241.533 30 236.942 272.376 310.991 31 243.452 278.668 317.283 32 243.838 278.850 317.399 33 257.059 295.172 336.935 34 257.827 295.377 336.992 35 258.707 295.621 337.061 36 271.319 307.196 347.768 37 272.845 307.760 348.071 38 275.872 311.499 352.360 39 318.108 363.792 414.834 40 318.854 364.008 414.896 41 319.831 364.299 414.979 42 324.453 368.756 419.262 43 324.985 368.898 419.354 44 329.826 373.311 423.534 45 379.328 432.449 492.741 46 381.553 433.165 492.952 47 382.991 433.562 493.062 48 388.570 439.965 499.765 49 390.352 440.522 499.981 50 407.686 454.706 512.499 51 422.016 478.719 544.819 52 423.208 479.144 544.948 53 423.951 479.299 544.986 54 427.951 480.424 545.305 55 446.064 499.430 563.915 56 451.676 501.353 564.718 57 506.050 571.046 648.928 58 507.530 571.641 649.126 59 510.351 572.493 649.361 60 513.073 576.012 653.197

2 2

2 3

2 4

3 3

3 4

4 3

4 4

1.000 1.000 4.657 4.657 4.657 7.037 7.037 8.435 76.680 76.680 76.680 76.680 76.680 76.680 81.985 85.682 85.682 92.709 92.709 92.709 106.121 106.121 110.638 110.638 110.638 110.638 233.558 240.589 240.589 264.016 264.016 264.016 278.894 278.894 288.345 313.804 313.804 313.804 313.804 313.804 313.804 333.632 338.157 338.157 343.676 343.676 343.676 349.576 349.576 350.862 350.862 350.862 350.862

1.000 1.000 4.604 4.604 4.604 6.897 6.897 8.221 90.646 90.646 90.646 90.646 90.646 90.646 95.727 99.281 99.281 106.029 106.029 106.029 118.696 118.696 122.844 122.844 122.844 122.844 272.973 279.670 279.670 302.510 302.510 302.510 317.203 317.203 326.492 364.659 364.659 364.659 364.659 364.659 364.659 378.399 383.320 383.320 389.671 389.671 389.671 396.889 396.889 398.526 398.526 398.526 398.526 641.800 641.800 641.800 641.800 641.800 641.800

1.000 1.000 4.564 4.564 4.564 6.796 6.796 8.069 104.909 104.909 104.909 104.909 104.909 104.909 109.862 113.334 113.334 119.916 119.916 119.916 132.143 132.143 136.080 136.080 136.080 136.080 315.026 321.593 321.593 344.223 344.223 344.223 358.719 358.719 367.669 420.234 420.234 420.234 420.234 420.234 420.234 431.163 436.025 436.025 442.707 442.707 442.707 450.756 450.756 452.644 452.644 452.644 452.644 736.460 736.460 736.460 736.460 736.460 736.460 746.830

1.000 1.000 4.935 4.935 4.935 7.914 7.914 9.986 22.861 22.861 22.861 22.861 22.861 22.861 28.189 30.667 30.667 34.410 34.410 34.410 39.717 39.717 41.138 41.138 41.138 41.138 406.713 406.713 406.713 406.713 406.713 406.713 406.713 406.713 406.713 406.713 406.713 406.713 406.713 406.713 406.713 432.341 432.341 432.341 432.341 436.018 436.018 450.234 450.234 450.234 460.759 460.759 467.972 484.227 484.227 484.227 484.227 484.227 484.227 528.618

1.000 1.000 4.925 4.925 4.925 7.886 7.886 9.939 22.586 22.586 22.586 22.586 22.586 22.586 27.758 30.149 30.149 33.744 33.744 33.744 38.804 38.804 40.150 40.150 40.150 40.150 475.493 475.493 475.493 475.493 475.493 475.493 475.493 475.493 475.493 475.493 475.493 475.493 475.493 475.493 475.493 500.222 500.222 500.222 500.222 503.779 503.779 517.537 517.537 517.537 527.721 527.721 534.698 550.392 550.392 550.392 550.392 550.392 550.392 592.902

1.000 1.000 4.990 4.990 4.990 8.075 8.075 10.257 24.692 24.692 24.692 24.692 24.692 24.692 31.278 34.517 34.517 39.682 39.682 39.682 47.772 47.772 50.148 50.148 50.148 50.148 116.005 116.005 116.005 116.005 116.005 116.005 116.005 116.005 116.005 116.005 116.005 116.005 116.005 116.005 116.005 143.736 143.736 143.736 143.736 146.633 146.633 156.739 156.739 156.739 163.397 163.397 167.656 176.528 176.528 176.528 176.528 176.528 176.528 197.033

1.000 1.000 4.987 4.987 4.987 8.067 8.067 10.244 24.612 24.612 24.612 24.612 24.612 24.612 31.148 34.357 34.357 39.469 39.469 39.469 47.459 47.459 49.802 49.802 49.802 49.802 114.013 114.013 114.013 114.013 114.013 114.013 114.013 114.013 114.013 114.013 114.013 114.013 114.013 114.013 114.013 140.589 140.589 140.589 140.589 143.347 143.347 152.945 152.945 152.945 159.245 159.245 163.265 171.614 171.614 171.614 171.614 171.614 171.614 190.775

Table 3.4: Sierpinski Gasket, No Overlap, Normalized 20

4

Non-PCF Fractals

Our first example is the octagasket, generated by eight contractive homotheties √ with contraction ratio 1 − 2/2 and fixed points {qi } the vertices of a regular octagon. Then the consecutive images Fi K and Fi+1 K intersect along a Cantor set. As yet, there has been no construction of a self-similar Laplacian on this fractal, although it is reasonable to expect the probabilistic methods in [Barlow 1995] will work, given the high symmetry in this example. It is natural to approximate from without by taking Ω to be the interior of the octagon with vertices {qi }. Then Ωm consists of the interior of the union of 8m octagons that meet along edges. In table 4.1 we give the eigenvalues on Ωm for m = 0, 1, 2, 3 along with levelto-level ratios, suggesting a renormalization factor of about r = 1.2. In Table (m) 4.2 we normalize the eigenvalues by dividing by λ1 . This suggests an eigenvalue renormalization factor of about R = 14.9476 (the table indicates when a new eigenvalue appears that is approximately Rλn for an earlier value of n). In the next section we will explain why this happens. The tables show eigenvalues of multiplicities 1 and 2, but no higher multiplicities. The D8 symmetry forces multiplicity 2, since there are three irreducible representations of dimension 2. There are a number of close coincidences (for example 910.5058 and 910.8645, each with multiplicity 2), but not close enough to be regarded as the same, in our judgement. There is some evidence of large gaps in the spectrum, for example (66.45202,122.0411), (162.1709,223.2267) and (253.6123,336.1848). However, there is not enough data to guess whether or not there are infinitely many gaps (λj+1 /λj ≥ 1 + for fixed ). In Figure 4.1 we display the graphs of some eigenfunctions, and in Figure 4.2 we show the Weyl ratios. The Weyl ratio is defined to be W (x) = N (x)/xα , where N (x) = #{λj ≤ x} is the eigenvalue counting function, and xα is its approximate growth rate. We determine α experimentally as the slope of the line of best fit to a log-log plot of N (x). The Weyl ratio gives a nice “snapshot” of the spectrum. A question of interest is whether it tends to a limit, or shows periodic behavior for large x. Our experimental data does not give an indication of what answer to expect. The next example we consider is the standard SC generated by eight contractions of ratio 13 (omitting the middle tic-tac-toe square). Here the existence of a self-similar Laplacian is known, and as stated above, uniqueness is established in [Barlow et al. 2008]. Here it is natural to choose Ω to be the interior of the square that just contains SC, so Ωm contains 8m squares of side length 3−m intersecting along edges. In Tables 4.3 and 4.4 we report unnormalized and normalized eigenvalue data, as before. In Table 4.5 we describe the D4 representation type associated to the eigenspace. There is one 2-dimensional representation (denoted 2) and four 1-dimensional representations (1 + +, 1 + −, 1 − +, and 1 − −) described in more detail in the next section. Again we only see eigenvalue multiplicities of 1 or 2. There is an apparent eigenvalue renormalization factor of about R = 10.0081, which is consistent with computations in [Barlow et al. 1990]. In the next section we will give an explanation of this behavior. Spectral gaps are consistent with the data. Figure 4.3 shows some 21

Octagasket Unnormalized Eigenvalues

Level: Refinement: n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 58 59 60

Ratios (j) (j+1) λn /λn , highest refinements used

1 0

1 1

2 1

2 2

2 3

2 4

3 1

3 2

3 3

3 4

4 1

12.87 12.87 35.55 35.56 57.06 67.47 67.49 99.03 112.63 112.78 124.95 166.13 166.21 179.85 180.29 201.34 237.11 237.65 274.23 274.43 289.57 290.74 335.78 336.40 360.47 379.58 400.07 428.17 441.65 445.04 486.74 500.94 503.98 517.82 519.43 607.09 611.05 638.35 644.93 659.58 665.39 669.03 671.06 752.63 799.56 802.16 815.03 832.53 840.23 870.15 895.83 904.07 951.97 977.47 983.56 1008.08 1035.98 1059.87 1131.15

12.81 12.81 35.14 35.15 55.93 66.00 66.00 95.78 108.51 108.55 120.29 157.78 157.80 169.80 169.87 188.62 220.91 221.00 251.90 251.95 265.40 265.65 304.52 304.62 322.62 339.03 356.68 381.56 390.22 390.90 425.77 434.79 435.36 450.69 451.06 518.07 518.98 538.29 539.72 557.54 558.99 569.71 569.92 614.76 664.07 664.43 679.96 685.05 686.13 712.75 718.89 722.76 779.21 790.31 791.59 813.30 828.30 841.04 901.70

6.28 6.30 23.98 24.00 47.97 48.24 62.46 151.54 151.61 154.94 155.27 161.09 161.44 162.62 163.28 223.56 225.86 227.27 251.17 251.31 290.09 292.73 314.75 428.94 431.35 448.72 452.24 453.90 459.24 479.16 488.40 562.07 565.73 571.43 573.83 584.89 643.32 651.28 700.98 749.55 759.91 764.98 768.33 780.01 790.04 792.39 806.00 847.00 849.40 889.09 899.51 951.70 956.27 984.78 987.21 1099.26 1101.07 1103.77 1117.52

6.14 6.15 23.37 23.38 46.60 46.70 60.28 149.80 149.81 152.84 152.92 158.27 158.37 159.42 159.63 213.46 217.24 217.60 242.41 242.44 278.88 279.78 299.70 413.83 414.35 432.14 432.96 433.38 434.84 456.52 458.88 535.94 537.94 539.01 539.58 549.09 605.90 607.91 661.74 703.59 706.48 710.86 711.94 716.52 724.17 727.47 740.88 785.98 786.33 821.77 824.78 867.27 869.23 894.52 895.16 990.90 996.03 999.31 1000.43

6.08 6.09 23.14 23.14 46.08 46.12 59.47 149.30 149.31 152.22 152.24 157.40 157.43 158.46 158.52 210.19 214.55 214.66 239.73 239.75 275.54 275.84 295.20 409.52 409.64 427.30 427.49 427.60 427.99 450.02 450.63 528.39 529.17 530.14 530.53 539.29 595.21 595.75 651.18 690.24 690.87 695.42 696.68 696.98 709.42 710.39 723.51 769.26 769.32 803.21 803.97 844.89 845.43 869.56 869.79 961.30 962.98 971.51 971.72

6.06 6.06 23.04 23.04 45.88 45.90 59.17 149.17 149.17 152.04 152.04 157.14 157.14 158.17 158.19 209.10 213.67 213.72 238.90 238.90 274.51 274.61 293.80 408.37 408.40 425.95 425.99 426.02 426.12 448.16 448.32 526.24 526.46 527.96 528.06 536.57 592.22 592.37 648.45 686.48 686.62 689.38 692.85 692.92 705.55 705.82 718.84 764.87 764.88 798.22 798.40 838.85 838.99 862.77 862.84 953.18 953.67 964.09 964.14

5.07 5.07 19.15 19.15 38.75 38.75 53.10 75.96 75.96 78.99 78.99 84.90 84.90 85.54 85.54 157.34 157.34 159.54 168.29 168.29 193.38 193.38 209.59 290.88 290.88 309.23 309.23 310.63 310.63 331.23 331.23 437.21 437.21 441.53 441.53 459.56 493.31 493.31 557.35 588.09 588.09 616.83 616.83 649.02 649.02 673.48 673.48 712.51 712.51 737.80 737.80 749.87 749.87 763.18 769.50 968.08 1005.76 1005.76 1078.51

4.86 4.86 18.37 18.37 37.15 37.15 50.89 72.75 72.75 75.65 75.65 81.30 81.30 81.91 81.91 150.33 150.33 152.43 160.76 160.76 184.60 184.60 199.98 277.07 277.07 294.38 294.38 295.70 295.70 315.14 315.14 414.29 414.29 418.44 418.44 435.33 466.56 466.56 525.93 554.32 554.32 580.78 580.78 610.27 610.27 632.54 632.54 668.11 668.11 690.37 690.37 701.93 701.93 713.92 718.12 909.14 941.74 941.74 1005.57

4.78 4.78 18.04 18.04 36.47 36.47 49.96 71.40 71.40 74.25 74.25 79.79 79.79 80.39 80.39 147.46 147.46 149.51 157.67 157.67 181.03 181.03 196.08 271.53 271.53 288.46 288.46 289.75 289.75 308.75 308.75 405.37 405.37 409.46 409.46 425.94 456.26 456.26 513.96 541.50 541.50 567.16 567.16 595.70 595.70 617.22 617.22 651.58 651.58 672.83 672.83 684.17 684.17 695.71 699.26 887.65 918.58 918.58 979.42

4.74 4.74 17.90 17.90 36.20 36.20 49.58 70.86 70.86 73.68 73.68 79.18 79.18 79.78 79.78 146.31 146.31 148.34 156.43 156.43 179.60 179.60 194.53 269.34 269.34 286.11 286.11 287.39 287.39 306.23 306.23 401.88 401.88 405.95 405.95 422.28 452.25 452.25 509.33 536.55 536.55 561.91 561.91 590.10 590.10 611.34 611.34 645.25 645.25 666.14 666.14 677.40 677.40 688.77 692.09 879.56 909.87 909.87 969.63

3.95 3.95 14.93 14.93 30.19 30.19 41.32 59.09 59.09 61.42 61.42 65.96 65.96 66.45 66.45 122.04 122.04 123.73 130.46 130.46 149.71 149.71 162.17 223.23 223.23 237.13 237.13 238.19 238.19 253.61 253.61 336.18 336.18 338.61 338.61 351.80 379.14 379.14 428.58 451.74 451.74 473.92 473.92 499.69 499.69 519.17 519.17 551.07 551.07 576.94 576.94 582.26 582.26 593.76 619.00 668.25 712.82 712.82 769.87

Table 4.1: Octagasket Unnormalized Eigenvalues and Ratios

22

2.11 2.11 1.53 1.53 1.22 1.44 1.12 0.64 0.73 0.71 0.79 1.00 1.00 1.07 1.07 0.90 1.03 1.03 1.05 1.05 0.97 0.97 1.04 0.75 0.79 0.80 0.84 0.90 0.92 0.87 0.95 0.83 0.83 0.85 0.85 0.97 0.88 0.91 0.83 0.81 0.81 0.83 0.82 0.89 0.94 0.94 0.95 0.90 0.90 0.89 0.90 0.86 0.93 0.92 0.92 0.85 0.87 0.87 0.94

1.28 1.28 1.29 1.29 1.27 1.27 1.19 2.11 2.11 2.06 2.06 1.98 1.98 1.98 1.98 1.43 1.46 1.44 1.53 1.53 1.53 1.53 1.51 1.52 1.52 1.49 1.49 1.48 1.48 1.46 1.46 1.31 1.31 1.30 1.30 1.27 1.31 1.31 1.27 1.28 1.28 1.23 1.23 1.17 1.20 1.15 1.18 1.19 1.19 1.20 1.20 1.24 1.24 1.25 1.25 1.08 1.05 1.06 0.99

1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.21 1.21 1.21 1.21 1.21 1.21 1.21 1.21 1.20 1.20 1.20 1.20 1.20 1.19 1.19 1.19 1.19 1.19 1.19 1.19 1.18 1.18 1.18 1.18 1.17 1.17 1.15 1.15 1.16 1.16 1.16 1.12 1.32 1.28 1.28 1.26

Octagasket Normalized Eigenvalues Level: 1 1 2 0 1 1 Refinement: n 1 1.00 1.00 1.00 2 1.00 1.00 1.00 2.76 2.74 3.82 3 4 2.76 2.74 3.82 4.43 4.37 7.64 5 6 5.24 5.15 7.68 5.25 5.15 9.95 7 8 7.70 7.48 24.13 9 8.75 8.47 24.14 10 8.77 8.47 24.67 11 9.71 9.39 24.72 25.65 12 12.91 12.31 13 12.92 12.32 25.70 25.89 14 13.98 13.25 15 14.01 13.26 26.00 35.59 16 15.65 14.72 17 18.43 17.24 35.96 18 18.47 17.25 36.18 39.99 19 21.31 19.66 20 21.33 19.66 40.01 21 22.51 20.71 46.19 46.61 22 22.60 20.73 23 26.10 23.77 50.11 24 26.15 23.78 68.29 25 28.02 25.18 68.68 26 29.50 26.46 71.44 27 31.09 27.84 72.00 72.27 28 33.28 29.78 29 34.33 30.46 73.12 30 34.59 30.51 76.29 31 37.83 33.23 77.76 32 38.93 33.93 89.49 33 39.17 33.98 90.07 34 40.25 35.18 90.98 35 40.37 35.20 91.36 36 47.18 40.43 93.12 37 47.49 40.51 102.42 38 49.61 42.01 103.69 39 50.13 42.12 111.60 40 51.26 43.52 119.34 41 51.72 43.63 120.99 42 52.00 44.47 121.79 43 52.16 44.48 122.33 44 58.50 47.98 124.19 45 62.14 51.83 125.78 46 62.35 51.86 126.16 47 63.34 53.07 128.32 48 64.70 53.47 134.85 49 65.30 53.55 135.23 50 67.63 55.63 141.55 51 69.62 56.11 143.21 52 70.27 56.41 151.52 53 73.99 60.82 152.25 54 75.97 61.68 156.79 55 76.44 61.78 157.17 56 78.35 63.48 175.01 57 80.52 64.65 175.30 58 82.37 65.64 175.73 59 87.91 70.38 177.92 60 88.87 70.49 195.56

2 2

2 3

2 4

3 1

3 2

3 3

3 4

4 1

1.00 1.00 3.81 3.81 7.59 7.60 9.82 24.39 24.39 24.89 24.90 25.77 25.79 25.96 25.99 34.76 35.37 35.43 39.47 39.48 45.41 45.56 48.80 67.39 67.47 70.37 70.50 70.57 70.81 74.34 74.72 87.27 87.59 87.77 87.86 89.41 98.66 98.99 107.75 114.57 115.04 115.75 115.93 116.67 117.92 118.46 120.64 127.98 128.04 133.81 134.30 141.22 141.54 145.66 145.76 161.35 162.19 162.72 162.90 174.63

1.00 1.00 3.80 3.80 7.57 7.58 9.77 24.54 24.54 25.02 25.02 25.87 25.87 26.04 26.05 34.54 35.26 35.28 39.40 39.40 45.28 45.33 48.51 67.30 67.32 70.22 70.26 70.27 70.34 73.96 74.06 86.84 86.96 87.12 87.19 88.63 97.82 97.91 107.02 113.44 113.54 114.29 114.49 114.54 116.59 116.75 118.90 126.42 126.43 132.00 132.13 138.85 138.94 142.90 142.94 157.98 158.26 159.66 159.69 169.25

1.00 1.00 3.80 3.80 7.57 7.57 9.76 24.61 24.61 25.08 25.08 25.92 25.92 26.09 26.09 34.49 35.25 35.25 39.41 39.41 45.28 45.30 48.46 67.36 67.37 70.26 70.27 70.27 70.29 73.92 73.95 86.80 86.84 87.09 87.10 88.51 97.69 97.71 106.96 113.23 113.26 113.71 114.29 114.30 116.38 116.42 118.57 126.16 126.17 131.67 131.70 138.37 138.39 142.31 142.32 157.23 157.31 159.03 159.03 167.91

1.00 1.00 3.78 3.78 7.64 7.64 10.48 14.99 14.99 15.58 15.58 16.75 16.75 16.88 16.88 31.04 31.04 31.47 33.20 33.20 38.15 38.15 41.35 57.38 57.38 61.00 61.00 61.28 61.28 65.34 65.34 86.25 86.25 87.10 87.10 90.66 97.32 97.32 109.95 116.01 116.01 121.68 121.68 128.03 128.03 132.86 132.86 140.56 140.56 145.55 145.55 147.93 147.93 150.55 151.80 190.97 198.41 198.41 212.76 212.76

1.00 1.00 3.78 3.78 7.64 7.64 10.46 14.96 14.96 15.55 15.55 16.71 16.71 16.84 16.84 30.91 30.91 31.34 33.05 33.05 37.95 37.95 41.11 56.96 56.96 60.52 60.52 60.79 60.79 64.79 64.79 85.17 85.17 86.02 86.02 89.50 95.92 95.92 108.12 113.96 113.96 119.40 119.40 125.46 125.46 130.04 130.04 137.35 137.35 141.93 141.93 144.30 144.30 146.77 147.63 186.90 193.61 193.61 206.73 206.73

1.00 1.00 3.78 3.78 7.64 7.64 10.46 14.95 14.95 15.54 15.54 16.70 16.70 16.83 16.83 30.87 30.87 31.30 33.01 33.01 37.90 37.90 41.05 56.84 56.84 60.39 60.39 60.66 60.66 64.64 64.64 84.86 84.86 85.72 85.72 89.17 95.52 95.52 107.60 113.36 113.36 118.73 118.73 124.71 124.71 129.21 129.21 136.40 136.40 140.85 140.85 143.23 143.23 145.64 146.39 185.83 192.30 192.30 205.04 205.04

1.00 1.00 3.78 3.78 7.64 7.64 10.46 14.95 14.95 15.54 15.54 16.70 16.70 16.83 16.83 30.86 30.86 31.29 32.99 32.99 37.88 37.88 41.03 56.81 56.81 60.35 60.35 60.62 60.62 64.59 64.59 84.76 84.76 85.62 85.62 89.07 95.39 95.39 107.43 113.17 113.17 118.52 118.52 124.46 124.46 128.94 128.94 136.09 136.09 140.50 140.50 142.87 142.87 145.27 145.97 185.51 191.91 191.91 204.51 204.51

1.00 1.00 3.78 3.78 7.64 7.64 10.45 14.95 14.95 15.54 15.54 16.69 16.69 16.81 16.81 30.87 30.87 31.30 33.00 33.00 37.87 37.87 41.02 56.47 56.47 59.98 59.98 60.25 60.25 64.15 64.15 85.04 85.04 85.65 85.65 88.99 95.91 95.91 108.41 114.27 114.27 119.88 119.88 126.40 126.40 131.33 131.33 139.40 139.40 145.94 145.94 147.29 147.29 150.20 156.58 169.04 180.32 180.32 194.75 194.75

Table 4.2: Octagasket Normalized Eigenvalues. Eigenvalues in boldface on level 4 are approximately R (14.95) times the eigenvalues in boldface on level 3

23

Figure 4.1: Octagasket Eigenfunctions, Level 3 24

Figure 4.2: Octagasket Weyl Ratios, Level 4, 1 Refinement, α = .71938 eigenfunctions and Figure 4.4 shows the Weyl ratios. The last two examples we consider are alternate carpets. We subdivide the unit square into 16 subsquares of side length 41 , and retain all but the 4 inner 13 12 squares ( 12 16 carpet) or all but 3 of the inner squares ( 16 carpet). The 16 carpet 13 has D4 symmetry and it is known that a self-similar Laplacian exists. The 16 carpet has no symmetry, and the methods used to construct a Laplacian on SC do not work on this example. So the situation is even worse than for the octagasket. In tables 4.6 and 4.7 we present unnormalized and normalized eigenvalues 12 for the 16 carpet, and in Tables 4.8 and 4.9 the same data for the 13 16 carpet. (Again we use the interior of the square for Ω). In Figure 4.5 we show the Weyl 13 ratios for the 12 16 carpet, and in Figure 4.6 we show those for the 16 carpet. The evidence for convergence is strong in both cases. But the nature of the 12 spectrum is quite different. In the symmetric 16 carpet, we see multiplicities of 1 or 2, and an eigenvalue renormalization factor of about R = 20.123. For the 13 16 carpet we do not see any multiplicities above 1, and there is no apparent eigenvalue renormalization factor. The evidence for spectral gaps is also weaker for the 13 16 carpet, but this is not conclusive.

25

SC Unnormalized Eigenvalue Data Level: 1 1 0 1 Refinement: n 1 0.0000 0.0000 2 6.9095 6.8043 6.9151 6.8070 3 4 18.9925 18.8600 5 34.3954 33.3153 47.0332 45.9328 6 7 47.1243 45.9546 52.1193 51.1387 8 9 92.8584 89.8444 93.0125 89.8865 10 11 99.0749 95.5391 99.2335 12 103.3472 13 116.5071 111.4375 14 138.4478 129.5749 15 139.6143 132.0182 16 140.8159 132.2970 17 176.7080 165.3371 18 177.1511 165.4399 19 186.3468 174.2256 20 196.9752 182.4076 21 207.4874 192.1334 22 255.1933 232.4075 23 258.1854 233.1381 24 281.0056 253.6079 25 294.4730 267.9779 26 296.0104 268.7574 27 299.1848 269.4117 28 331.3359 297.2510 29 375.1390 328.3141 30 411.4908 363.1051 31 415.5502 364.2623 32 422.8709 372.1289 33 428.1018 373.1788 34 436.6960 380.8205 35 439.9719 381.6613 36 451.1346 393.6453 37 483.1916 416.8176 38 493.4493 422.8265 39 517.9593 439.4004 40 521.0245 440.3721 41 554.2167 471.2427 42 561.4218 472.5244 43 605.5574 504.9343 44 616.5028 518.5646 45 628.0957 524.1471 46 634.2437 531.7247 47 641.5337 532.7246 48 667.4687 547.9734 49 698.1628 573.9857 50 700.7259 574.8348 51 752.4778 618.6779 52 772.0233 624.3244 53 818.5900 672.0718 54 833.2754 676.1826 55 846.5291 681.5713 56 861.5834 694.2754 57 868.5235 702.4581 58 869.8747 703.6618 59 874.0871 704.4428 60 969.8144 782.3749

1 2

1 3

1 4

2 0

2 1

2 2

2 3

3 0

3 1

3 2

3 3

0.0000 6.7653 6.7664 18.8243 32.9506 45.6120 45.6175 50.8862 89.0822 89.0933 94.6410 98.1700 110.1583 127.0433 130.0016 130.0662 162.4409 162.4692 171.1654 178.8404 188.2268 226.4589 226.6316 246.3701 261.2521 261.9548 262.0976 288.6697 316.2114 350.9457 351.2403 359.5109 359.7477 367.3222 367.5250 379.2330 400.4065 405.5962 419.9247 420.1850 450.8932 451.1818 479.8486 493.7396 498.5358 505.9813 506.1940 517.8253 543.1162 543.7180 584.2646 585.6444 631.6665 632.6191 641.2196 650.1452 659.4855 659.8472 659.9817 728.5923

0.0000 6.7505 6.7510 18.8150 32.8218 45.5142 45.5157 50.8221 88.8905 88.8933 94.4145 97.8905 109.8362 126.2799 129.4432 129.4575 161.6916 161.6997 170.3923 177.9499 187.2187 224.8390 224.8783 244.3644 259.5384 260.2517 260.2854 286.5081 312.9576 347.8623 347.9359 356.3576 356.4153 363.9749 364.0248 375.6239 396.3120 401.2680 414.9014 414.9666 445.8230 445.8931 473.4384 487.4812 492.2513 499.5675 499.6197 510.1141 535.4598 535.8214 575.3579 575.6898 621.5085 621.7422 631.1787 639.0805 648.8736 649.0127 649.0347 715.1103

0.0000 6.7449 6.7451 18.8127 32.7746 45.4829 45.4834 50.8059 88.8425 88.8432 94.3576 97.8154 109.7554 126.0367 129.2825 129.2854 161.4947 161.4971 170.1977 177.7272 186.9544 224.3803 224.3886 243.7824 259.1060 259.8246 259.8329 285.9648 312.0478 347.0751 347.0933 355.5688 355.5831 363.1377 363.1501 374.7192 395.2878 400.1626 413.5579 413.5741 444.5550 444.5724 471.7517 485.9106 490.6859 497.9617 497.9748 508.0784 533.5479 533.7649 572.9848 573.0624 618.9558 619.0143 628.6572 636.2458 646.2178 646.3092 646.3133 711.7366

0.0000 6.4313 6.4323 17.0488 32.1097 42.8446 42.8729 44.8902 66.1417 66.2982 71.9960 75.6967 90.1313 108.9329 109.1286 113.3883 161.7668 162.4681 168.6568 178.1670 185.7692 224.8288 225.7046 228.0738 267.3355 268.9097 287.3782 308.4704 327.3493 358.6256 366.8938 370.3870 402.0958 426.2507 431.0532 469.0618 481.9094 496.4015 504.2032 515.6273 536.0207 552.9508 558.1539 567.3412 574.2506 598.2522 608.7209 630.2738 664.4216 670.4290 683.4854 701.0202 803.3161 805.8113 832.7409 848.3769 866.9036 881.1987 885.7117 901.5093

0.0000 6.2251 6.2254 16.5574 30.7611 41.1639 41.1729 43.1147 62.7984 62.8473 68.3327 71.6324 85.5167 102.2554 102.3168 104.7730 152.1424 152.3584 155.5422 171.8544 174.5895 207.6838 211.4197 211.4581 245.9265 246.0942 265.0656 283.9907 299.1712 317.3075 332.0219 332.2614 361.9981 379.0875 380.0237 415.1885 429.2890 433.4022 436.6132 445.0090 478.3656 482.3672 482.9689 488.3777 492.2321 519.5769 521.6171 531.2947 562.5890 576.8794 579.9359 589.9260 678.8308 679.9199 709.9328 711.2532 733.0618 754.4762 755.2010 767.8076

0.0000 6.1354 6.1355 16.3441 30.2096 40.4941 40.4965 42.3923 61.4619 61.4753 66.9090 70.0686 83.8352 99.8578 99.8748 101.6793 148.7128 148.7706 151.3127 169.9944 170.8667 200.9126 207.1909 207.2012 238.5531 238.5881 258.6422 277.1404 290.6243 302.3844 321.4484 321.5045 350.5678 363.7003 363.9265 398.1174 414.4408 416.1715 416.8702 422.9119 461.6796 461.9683 463.2601 465.6960 468.0827 494.7583 495.0149 504.3023 529.0856 551.2608 551.3653 560.3374 641.9561 642.3952 668.0265 670.9435 700.4748 717.1605 718.9525 720.6769

0.0000 6.10 6.10 16.26 30.00 40.24 40.24 42.12 60.97 60.97 66.39 69.50 83.24 99.01 99.01 100.57 147.54 147.56 149.93 169.49 169.66 198.63 205.92 205.93 236.14 236.15 256.75 275.23 288.04 297.35 318.28 318.29 347.27 358.88 358.94 392.86 409.86 411.27 411.39 416.05 455.99 456.00 458.71 459.52 460.80 487.00 487.05 496.86 518.77 543.26 543.30 553.09 631.51 631.64 655.94 659.33 690.67 706.72 708.32 708.45

0.0000 5.87 5.87 15.57 28.99 38.71 38.71 40.48 58.99 58.99 64.16 67.31 80.14 96.14 96.14 98.96 141.59 141.59 145.60 156.72 161.04 193.66 194.18 194.18 226.58 226.58 242.91 256.13 273.39 296.49 302.48 302.48 328.13 346.65 346.65 378.00 390.66 394.69 394.69 404.24 412.24 425.16 425.16 434.77 454.40 466.46 466.46 473.55 503.18 514.10 514.10 514.95 585.09 590.98 592.82 592.82 610.41 610.41 611.55 611.66

0.0000 5.6310 5.6310 14.9425 27.7633 37.0938 37.0938 38.7875 56.4159 56.4159 61.3551 64.3456 76.6848 91.8300 91.8300 94.3752 135.1528 135.1528 138.8378 150.0903 153.7589 184.0351 185.2580 185.2580 215.4968 215.4968 231.2213 244.2915 259.6175 279.3908 286.9005 286.9005 311.5196 327.5608 327.5608 357.2698 369.4654 373.5203 373.5203 380.8049 390.5909 402.1807 402.1807 411.2604 428.3600 438.7119 438.7119 445.8369 473.2766 483.1208 483.1208 483.2307 547.0977 552.3832 554.2646 554.2646 569.6585 569.6585 570.8047 572.3659

0.0000 5.5325 5.5325 14.6841 27.2621 36.4341 36.4341 38.0972 55.3699 55.3699 60.2173 63.1414 75.2891 90.0930 90.0930 92.5219 132.6025 132.6025 136.1567 147.4883 150.8882 180.2588 181.7748 181.7748 211.1887 211.1887 226.7026 239.7481 254.3202 272.8143 280.9431 280.9431 305.2009 320.3207 320.3207 349.4391 361.4892 365.5479 365.5479 372.0055 382.4977 393.5960 393.5960 402.4804 418.6438 428.4159 428.4159 435.5522 462.2661 471.5134 471.6932 471.6932 533.1663 538.3562 540.1749 540.1749 554.8341 554.8341 555.9232 558.0957

0.0000 5.4936 5.4936 14.5823 27.0651 36.1754 36.1754 37.8262 54.9598 54.9598 59.7724 62.6712 74.7457 89.4182 89.4182 91.7984 131.6229 131.6229 135.1289 146.4968 149.7908 178.8197 180.4537 180.4537 209.5562 209.5562 225.0039 238.0328 252.3353 270.3526 278.7202 278.7202 302.8469 317.6237 317.6237 346.5403 358.5420 362.6043 362.6043 368.7444 379.4817 390.4050 390.4050 399.2239 415.0663 424.6321 424.6321 431.7830 458.2523 467.2072 467.5056 467.5056 528.0563 533.2348 535.0612 535.0612 549.3760 549.3760 550.4958 552.8390

Table 4.3: SC Unnormalized Eigenvalues

26

SC Normalized Eigenvalue Data Level: 1 1 Refinement: 0 1 n 1 0.0000 0.0000 2 1.0000 1.0000 1.0008 1.0004 3 4 2.7488 2.7718 4.9780 4.8962 5 6 6.8070 6.7505 6.8202 6.7537 7 8 7.5431 7.5156 13.4392 13.2040 9 10 13.4615 13.2102 11 14.3389 14.0409 12 14.9572 14.5839 13 16.8618 16.3774 20.0373 19.0430 14 15 20.2061 19.4020 20.3800 19.4430 16 17 25.5746 24.2988 25.6387 24.3139 18 19 26.9696 25.6051 28.5078 26.8075 20 21 30.0292 28.2369 22 36.9336 34.1558 23 37.3666 34.2631 24 40.6694 37.2715 25 42.6185 39.3834 26 42.8410 39.4979 27 43.3004 39.5941 28 47.9536 43.6855 29 54.2931 48.2507 30 59.5542 53.3637 31 60.1417 53.5338 32 61.2012 54.6899 33 61.9583 54.8442 34 63.2021 55.9673 35 63.6762 56.0909 36 65.2918 57.8521 37 69.9313 61.2576 38 71.4159 62.1407 39 74.9632 64.5765 40 75.4068 64.7193 41 80.2106 69.2562 42 81.2534 69.4446 43 87.6411 74.2077 44 89.2252 76.2108 45 90.9030 77.0313 46 91.7928 78.1449 47 92.8479 78.2919 48 96.6014 80.5329 49 101.0437 84.3558 50 101.4146 84.4806 51 108.9046 90.9240 52 111.7333 91.7538 53 118.4728 98.7710 54 120.5982 99.3752 55 122.5164 100.1671 56 124.6952 102.0342 57 125.6996 103.2367 58 125.8952 103.4137 59 126.5048 103.5284 60 140.3592 114.9817

1 2

1 3

1 4

2 0

2 1

2 2

2 3

3 0

3 1

3 2

3 3

0.0000 1.0000 1.0002 2.7825 4.8706 6.7421 6.7429 7.5217 13.1676 13.1692 13.9892 14.5109 16.2829 18.7787 19.2160 19.2256 24.0110 24.0152 25.3006 26.4351 27.8225 33.4737 33.4993 36.4169 38.6166 38.7205 38.7416 42.6693 46.7404 51.8746 51.9181 53.1406 53.1756 54.2952 54.3252 56.0558 59.1856 59.9527 62.0706 62.1091 66.6482 66.6908 70.9282 72.9815 73.6904 74.7910 74.8224 76.5417 80.2800 80.3690 86.3623 86.5663 93.3690 93.5097 94.7810 96.1004 97.4810 97.5344 97.5543 107.6959

0.0000 1.0000 1.0001 2.7872 4.8621 6.7423 6.7425 7.5286 13.1679 13.1684 13.9862 14.5012 16.2708 18.7067 19.1753 19.1774 23.9524 23.9536 25.2413 26.3609 27.7340 33.3069 33.3127 36.1993 38.4471 38.5528 38.5578 42.4424 46.3605 51.5312 51.5421 52.7896 52.7982 53.9180 53.9254 55.6437 58.7083 59.4425 61.4621 61.4718 66.0427 66.0531 70.1336 72.2138 72.9205 74.0043 74.0120 75.5666 79.3212 79.3748 85.2316 85.2808 92.0682 92.1028 93.5007 94.6712 96.1220 96.1426 96.1458 105.9340

0.0000 1.0000 1.0000 2.7892 4.8592 6.7433 6.7434 7.5325 13.1719 13.1720 13.9895 14.5022 16.2724 18.6863 19.1675 19.1680 23.9433 23.9437 25.2337 26.3500 27.7180 33.2668 33.2680 36.1434 38.4153 38.5218 38.5230 42.3974 46.2645 51.4576 51.4603 52.7169 52.7190 53.8391 53.8409 55.5562 58.6057 59.3284 61.3144 61.3168 65.9101 65.9127 69.9423 72.0415 72.7495 73.8282 73.8301 75.3281 79.1042 79.1364 84.9512 84.9627 91.7669 91.7756 93.2052 94.3303 95.8088 95.8223 95.8229 105.5226

0.0000 1.0000 1.0002 2.6509 4.9927 6.6619 6.6663 6.9800 10.2844 10.3087 11.1947 11.7701 14.0146 16.9380 16.9684 17.6308 25.1532 25.2622 26.2245 27.7032 28.8853 34.9587 35.0949 35.4633 41.5681 41.8128 44.6845 47.9642 50.8997 55.7628 57.0484 57.5916 62.5220 66.2778 67.0246 72.9346 74.9322 77.1856 78.3987 80.1751 83.3460 85.9785 86.7875 88.2161 89.2904 93.0224 94.6502 98.0015 103.3111 104.2452 106.2754 109.0018 124.9079 125.2959 129.4831 131.9144 134.7951 137.0179 137.7196 140.1760

0.0000 1.0000 1.0001 2.6598 4.9414 6.6126 6.6140 6.9259 10.0879 10.0958 10.9769 11.5070 13.7374 16.4263 16.4361 16.8307 24.4401 24.4748 24.9863 27.6067 28.0460 33.3623 33.9624 33.9686 39.5056 39.5325 42.5801 45.6202 48.0588 50.9722 53.3359 53.3744 58.1513 60.8965 61.0469 66.6958 68.9609 69.6217 70.1375 71.4862 76.8446 77.4874 77.5840 78.4529 79.0721 83.4647 83.7925 85.3471 90.3742 92.6698 93.1608 94.7656 109.0473 109.2222 114.0435 114.2556 117.7589 121.1989 121.3154 123.3405

0.0000 1.0000 1.0000 2.6639 4.9238 6.6001 6.6005 6.9095 10.0176 10.0198 10.9055 11.4205 13.6643 16.2758 16.2786 16.5727 24.2386 24.2481 24.6624 27.7073 27.8495 32.7467 33.7700 33.7716 38.8817 38.8874 42.1560 45.1710 47.3687 49.2855 52.3928 52.4019 57.1389 59.2794 59.3162 64.8890 67.5495 67.8316 67.9455 68.9302 75.2490 75.2960 75.5066 75.9036 76.2926 80.6405 80.6823 82.1960 86.2354 89.8498 89.8668 91.3292 104.6322 104.7037 108.8814 109.3568 114.1701 116.8897 117.1818 117.4628

0.0000 1.0000 1.0000 2.6657 4.9181 6.5973 6.5974 6.9052 9.9946 9.9951 10.8833 11.3931 13.6459 16.2307 16.2314 16.4879 24.1877 24.1901 24.5786 27.7851 27.8134 32.5627 33.7589 33.7593 38.7131 38.7146 42.0914 45.1209 47.2201 48.7473 52.1778 52.1804 56.9307 58.8346 58.8439 64.4045 67.1924 67.4236 67.4430 68.2071 74.7537 74.7554 75.2006 75.3333 75.5423 79.8375 79.8453 81.4535 85.0454 89.0616 89.0675 90.6730 103.5291 103.5495 107.5333 108.0895 113.2266 115.8587 116.1207 116.1421

0.0000 1.0000 1.0000 2.6530 4.9383 6.5950 6.5950 6.8963 10.0497 10.0497 10.9307 11.4684 13.6534 16.3794 16.3794 16.8602 24.1225 24.1225 24.8067 26.7006 27.4371 32.9950 33.0824 33.0824 38.6022 38.6022 41.3849 43.6369 46.5777 50.5131 51.5339 51.5339 55.9042 59.0590 59.0590 64.4013 66.5581 67.2439 67.2439 68.8714 70.2347 72.4361 72.4361 74.0737 77.4183 79.4725 79.4725 80.6806 85.7282 87.5885 87.5885 87.7330 99.6834 100.6863 101.0009 101.0009 103.9967 103.9967 104.1921 104.2096

0.0000 1.0000 1.0000 2.6536 4.9305 6.5875 6.5875 6.8882 10.0189 10.0189 10.8960 11.4271 13.6184 16.3081 16.3081 16.7600 24.0017 24.0017 24.6561 26.6545 27.3060 32.6827 32.8999 32.8999 38.2700 38.2700 41.0625 43.3836 46.1053 49.6169 50.9505 50.9505 55.3226 58.1714 58.1714 63.4474 65.6132 66.3333 66.3333 67.6269 69.3648 71.4231 71.4231 73.0355 76.0722 77.9106 77.9106 79.1759 84.0489 85.7972 85.7972 85.8167 97.1588 98.0975 98.4316 98.4316 101.1654 101.1654 101.3689 101.6462

0.0000 1.0000 1.0000 2.6541 4.9276 6.5855 6.5855 6.8861 10.0081 10.0081 10.8843 11.4128 13.6085 16.2843 16.2843 16.7233 23.9679 23.9679 24.6103 26.6585 27.2730 32.5818 32.8558 32.8558 38.1724 38.1724 40.9765 43.3345 45.9684 49.3112 50.7805 50.7805 55.1651 57.8980 57.8980 63.1611 65.3392 66.0728 66.0728 67.2400 69.1365 71.1425 71.1425 72.7483 75.6699 77.4362 77.4362 78.7261 83.5546 85.2261 85.2586 85.2586 96.3698 97.3079 97.6366 97.6366 100.2863 100.2863 100.4831 100.8758

0.0000 1.0000 1.0000 2.6544 4.9267 6.5850 6.5850 6.8855 10.0043 10.0043 10.8804 11.4080 13.6059 16.2768 16.2768 16.7100 23.9593 23.9593 24.5975 26.6668 27.2664 32.5505 32.8479 32.8479 38.1455 38.1455 40.9574 43.3291 45.9325 49.2122 50.7354 50.7354 55.1271 57.8170 57.8170 63.0807 65.2653 66.0048 66.0048 67.1225 69.0770 71.0653 71.0653 72.6706 75.5544 77.2957 77.2957 78.5974 83.4156 85.0456 85.0999 85.0999 96.1220 97.0646 97.3971 97.3971 100.0028 100.0028 100.2066 100.6332

Table 4.4: SC Normalized Eigenvalues

27

Figure 4.3: Sierpinski Carpet (SC) Eigenfunctions, Level 4 28

Figure 4.4: SC Weyl Ratios, Level 3, 3 Refinements, α = .87392

Figure 4.5:

12 16

Carpet Weyl Ratios, Level 3, 0 Refinements, α = .71738

29

Sierpinski Carpet, Level 3, 3 Refinements, D4 Representation Type Number Eigenvalue Eigenfunction Number Eigenvalue Eigenfunction Type Type 1 5.4936 2 48 458.2523 1+ 2 5.4936 2 49 467.2072 1- + 3 14.5823 1+ 50 467.5056 2 51 467.5056 2 4 27.0651 1- + 5 36.1754 2 52 528.0563 1- + 6 36.1754 2 53 533.2348 1+ + 54 535.0612 2 7 37.8262 1+ + 8 54.9598 2 55 535.0612 2 56 549.3760 2 9 54.9598 2 10 59.7724 1+ 57 549.3760 2 58 550.4958 1+ 11 62.6712 1- 12 74.7457 1+ + 59 552.8390 1+ + 60 553.6998 2 13 89.4182 2 14 89.4182 2 15 91.7984 1- + 16 131.6229 2 17 131.6229 2 18 135.1289 1- + 19 146.4968 1+ 20 149.7908 1+ + 21 178.8197 1- 22 180.4537 2 23 180.4537 2 24 209.5562 2 25 209.5562 2 26 225.0039 1+ + 27 238.0328 1+ 28 252.3353 1- 29 270.3526 1- + 30 278.7202 2 31 278.7202 2 32 302.8469 1+ 33 317.6237 2 34 317.6237 2 35 346.5403 1+ + 36 358.5420 1- + 37 362.6043 2 38 362.6043 2 39 368.7444 1- 40 379.4817 1+ + 41 390.4050 2 42 390.4050 2 43 399.2239 1+ 44 415.0663 1+ + 45 424.6321 2 46 424.6321 2 47 431.7830 1- -

Table 4.5: D4 Representation Type

30

12/16 Symmetric Carpet Unnormalized Eigenvalues Level: 1 1 1 2 Refinement: 0 1 2 0 n 1 0.000 0.000 0.000 0.000 5.184 5.108 5.079 4.246 2 3 5.194 5.113 5.081 4.246 16.788 16.699 16.673 13.176 4 5 25.029 24.128 23.805 20.638 43.231 42.181 41.846 33.584 6 7 43.301 42.210 41.858 33.584 58.008 56.942 56.648 42.636 8 9 93.773 89.292 87.948 70.623 10 101.704 98.004 96.965 70.623 98.095 96.985 70.890 11 102.074 12 107.886 104.162 103.184 75.068 13 168.673 160.716 158.630 86.447 14 168.747 160.725 158.632 86.472 92.260 15 175.179 166.385 164.087 16 175.369 166.449 164.106 92.260 17 195.384 183.514 180.284 112.216 18 195.913 183.653 180.315 115.148 19 198.538 186.390 183.291 115.148 20 226.387 208.148 203.133 132.107 21 250.061 230.500 225.375 159.110 22 258.904 237.074 231.163 183.241 23 275.083 252.049 245.674 183.241 24 276.210 252.346 245.734 199.505 25 293.995 270.945 264.860 251.553 26 301.675 274.485 267.363 251.553 27 303.099 274.895 267.479 253.119 28 357.036 327.150 318.681 274.092 29 390.429 332.905 318.781 276.411 30 411.828 361.426 347.970 299.289 31 414.711 366.359 353.561 318.345 32 416.004 366.831 353.679 318.345 33 434.933 389.600 377.634 341.808 34 468.215 408.316 392.905 341.808 35 476.368 409.745 393.225 382.105 36 540.090 462.398 442.531 387.080 36 588.083 490.445 465.059 396.599 37 598.813 516.054 493.601 410.635 38 605.365 517.742 493.954 437.436 40 619.523 521.657 496.133 437.436 41 621.643 532.628 510.347 461.115 42 677.697 568.112 538.900 479.305 43 685.080 569.402 539.138 479.305 44 744.709 619.345 585.659 525.984 45 806.313 678.293 641.135 611.631 46 817.160 681.832 643.705 622.996 47 825.398 684.435 644.671 623.144 48 830.539 688.865 650.019 623.144 49 842.319 689.219 650.609 688.848 50 856.140 694.608 653.557 699.940 51 864.861 696.009 654.689 699.940 52 882.370 724.078 681.934 716.995 53 899.459 729.338 683.412 752.776 54 913.961 730.414 685.271 759.692 55 949.231 751.437 701.637 759.692 56 957.747 754.911 702.461 766.267 57 802.767 748.028 771.554 58 804.804 748.705 782.463 59 821.999 758.507 782.463 60 839.997 780.140 789.713

2 1

2 2

3 0

0.000 4.119 4.119 12.805 19.903 32.459 32.459 41.255 68.005 68.005 68.023 72.356 83.122 83.124 88.575 88.575 107.801 110.400 110.400 125.817 152.124 173.226 173.226 186.143 235.159 235.159 235.612 255.344 269.536 275.422 301.347 301.347 315.583 315.583 347.140 356.422 366.152 385.344 389.213 389.213 405.563 446.788 446.788 488.838 562.499 575.895 575.895 580.403 646.061 653.837 653.837 663.525 692.509 694.576 694.576 699.657 700.249 709.971 709.971 715.217

0.000 4.065 4.065 12.649 19.593 31.992 31.992 40.689 66.881 66.961 66.961 71.284 81.802 81.803 87.127 87.127 106.110 108.576 108.576 123.405 149.550 169.527 169.527 181.305 229.326 229.326 229.495 248.670 267.090 267.421 294.251 294.251 308.149 308.149 334.977 342.840 359.525 373.006 373.006 377.237 387.438 436.606 436.606 477.254 546.976 561.102 561.102 566.833 633.294 639.759 639.759 646.620 673.230 673.230 674.760 677.267 677.322 687.690 687.690 692.308

0.000 3.38 3.38 10.47 16.37 26.58 26.58 33.70 55.56 55.56 55.62 59.05 67.90 67.90 72.27 72.27 87.76 90.08 90.08 102.89 123.50 141.49 141.49 153.30 189.94 189.94 191.29 205.68 210.72 223.12 239.50 239.50 253.11 253.11 279.87 288.43 288.50 304.18 315.47 315.47 329.91 350.60 350.60 381.79 437.50 445.75 445.75 447.11 494.74 501.09 501.09 509.19 532.66 532.94 532.94 536.83 536.83 544.75 544.75 548.14

Table 4.6: 12/16 Carpet Unnormalized Eigenvalues

31

12/16 Symmetric Carpet Normalized Eigenvalues Level: 1 1 1 2 0 1 2 0 Refinement: n 1 0.000 0.000 0.000 0.000 2 1.000 1.000 1.000 1.000 3 1.002 1.001 1.000 1.000 4 3.239 3.269 3.283 3.103 5 4.829 4.724 4.687 4.861 8.340 8.258 8.239 7.910 6 7 8.354 8.263 8.242 7.910 11.191 11.147 11.154 10.042 8 9 18.091 17.481 17.317 16.633 19.621 19.186 19.092 16.633 10 11 19.692 19.204 19.096 16.696 20.813 20.392 20.316 17.680 12 13 32.540 31.463 31.233 20.360 14 32.555 31.465 31.234 20.366 15 33.795 32.573 32.308 21.729 16 33.832 32.586 32.312 21.729 37.693 35.927 35.497 26.429 17 18 37.795 35.954 35.503 27.120 38.302 36.490 36.089 27.120 19 20 43.674 40.749 39.996 31.114 21 48.242 45.125 44.375 37.474 22 49.948 46.412 45.515 43.157 23 53.069 49.344 48.372 43.157 24 53.286 49.402 48.384 46.988 25 56.717 53.043 52.150 59.246 26 58.199 53.736 52.642 59.246 27 58.474 53.816 52.665 59.615 28 68.879 64.046 62.747 64.554 29 75.321 65.173 62.766 65.101 30 79.450 70.756 68.514 70.489 31 80.006 71.722 69.614 74.977 32 80.255 71.815 69.638 74.977 33 83.907 76.272 74.354 80.503 34 90.328 79.936 77.361 80.503 35 91.901 80.216 77.424 89.993 36 104.194 90.524 87.132 91.165 37 113.453 96.015 91.568 93.407 38 115.523 101.028 97.188 96.713 39 116.787 101.359 97.257 103.025 40 119.518 102.125 97.686 103.025 41 119.927 104.273 100.485 108.602 42 130.741 111.219 106.107 112.886 43 132.165 111.472 106.154 112.886 44 143.669 121.249 115.313 123.880 45 155.554 132.790 126.236 144.052 46 157.646 133.482 126.742 146.728 47 159.236 133.992 126.933 146.763 48 160.227 134.859 127.986 146.763 49 162.500 134.929 128.102 162.238 50 165.166 135.984 128.682 164.850 51 166.849 136.258 128.905 164.850 52 170.226 141.753 134.269 168.867 53 173.523 142.783 134.560 177.294 54 176.321 142.993 134.927 178.923 55 183.125 147.109 138.149 178.923 56 184.768 147.789 138.311 180.472 57 157.158 147.283 181.717 58 157.557 147.416 184.286 59 160.923 149.346 184.286 60 164.446 153.606 185.994

2 1

2 2

3 0

0.000 1.000 1.000 3.109 4.831 7.879 7.879 10.015 16.508 16.508 16.513 17.565 20.178 20.178 21.502 21.502 26.169 26.800 26.800 30.542 36.928 42.051 42.051 45.186 57.085 57.085 57.195 61.985 65.430 66.859 73.152 73.152 76.608 76.608 84.268 86.521 88.884 93.542 94.482 94.482 98.451 108.458 108.458 118.666 136.547 139.799 139.799 140.893 156.832 158.719 158.719 161.071 168.107 168.609 168.609 169.842 169.986 172.346 172.346 173.619

0.000 1.000 1.000 3.112 4.820 7.870 7.870 10.009 16.453 16.472 16.472 17.536 20.123 20.124 21.433 21.433 26.103 26.710 26.710 30.358 36.789 41.704 41.704 44.601 56.414 56.414 56.456 61.173 65.704 65.786 72.386 72.386 75.805 75.805 82.405 84.339 88.443 91.760 91.760 92.801 95.310 107.405 107.405 117.405 134.556 138.032 138.032 139.441 155.791 157.381 157.381 159.069 165.615 165.615 165.991 166.608 166.622 169.172 169.172 170.308

0.000 1.000 1.000 3.097 4.841 7.859 7.859 9.967 16.431 16.431 16.449 17.463 20.081 20.081 21.373 21.373 25.953 26.640 26.640 30.428 36.523 41.843 41.843 45.338 56.173 56.173 56.572 60.827 62.317 65.987 70.830 70.830 74.855 74.855 82.767 85.300 85.321 89.958 93.297 93.297 97.568 103.688 103.688 112.911 129.385 131.825 131.825 132.228 146.313 148.192 148.192 150.587 157.527 157.610 157.610 158.761 158.761 161.104 161.104 162.107

Table 4.7: 12/16 Carpet Normalized Eigenvalues

32

13/16 Alternate Carpet Level: 1 0 Refinement: n 1 0.000 2 8.141 3 8.328 4 18.904 5 36.040 41.026 6 7 47.908 51.147 8 9 78.834 83.837 10 11 98.313 12 101.603 13 110.047 14 130.410 15 158.249 16 168.441 17 168.531 18 177.769 19 180.457 20 212.057 21 218.578 22 224.839 23 244.328 24 276.284 25 279.689 26 282.172 27 284.558 28 296.378 29 315.919 30 345.168 31 356.577 32 387.016 33 412.864 34 419.313 35 429.690 36 431.820 37 449.266 38 473.522 39 476.703 40 493.685 41 519.345 42 543.384 43 585.649 44 591.644 45 601.176 46 620.159 47 622.937 48 651.834 49 660.203 50 675.397 51 694.404 52 753.195 53 771.635 54 804.767 55 811.516 56 836.030 57 844.566 58 850.130 59 866.855 60 873.875

Unnormalized Eigenvalues 1 1 2 1 2 0 0.000 8.025 8.201 18.728 35.189 40.392 46.956 50.262 76.085 79.862 95.303 97.886 105.754 123.766 149.670 160.642 160.653 168.614 170.836 199.474 203.366 208.408 220.798 252.641 253.095 260.464 261.350 271.155 285.291 315.927 326.898 335.984 370.586 374.241 377.769 383.436 398.314 412.916 413.305 424.342 450.491 466.699 501.099 514.912 517.679 520.471 532.290 544.330 555.040 567.572 583.198 627.341 629.002 672.510 677.324 680.492 691.307 694.214 704.415 716.879

0.000 7.981 8.151 18.674 34.898 40.209 46.686 50.022 75.308 78.600 94.500 96.879 104.599 121.928 147.378 158.609 158.611 166.217 168.310 196.150 199.309 204.094 214.229 245.740 246.152 254.842 255.511 264.532 277.333 308.136 318.697 322.765 359.404 362.670 364.163 371.330 384.230 397.013 397.076 407.726 432.843 447.185 478.186 489.882 496.841 497.666 509.206 515.901 530.133 538.424 554.867 589.310 595.643 629.396 643.350 644.141 652.911 653.524 661.306 674.548

0.000 7.777 7.920 18.046 33.783 38.618 45.136 47.827 72.746 77.387 91.043 93.070 98.936 118.044 137.776 138.092 140.785 145.902 147.908 176.061 180.743 187.949 202.900 223.491 227.445 254.339 254.348 266.098 269.589 306.237 309.963 327.873 355.748 356.922 361.714 372.504 379.569 385.570 385.668 419.014 429.524 456.574 467.753 491.355 498.651 506.206 509.421 513.937 539.091 541.151 557.873 604.534 617.337 631.441 637.710 660.452 663.900 669.942 685.466 686.218

2 1

2 2

3 0

0.000 7.590 7.724 17.656 32.859 37.681 44.028 46.543 70.651 74.736 88.502 90.247 95.459 113.854 132.030 132.224 135.943 139.753 141.626 168.656 173.034 180.161 192.952 212.361 216.001 244.861 245.378 253.406 258.570 287.886 300.347 315.925 337.374 341.061 344.265 353.870 356.686 365.766 368.320 399.057 414.512 433.048 439.744 464.059 470.180 476.228 485.929 489.455 507.854 512.129 526.650 567.907 576.543 587.316 596.322 620.202 621.142 629.911 639.886 640.602

0.000 7.512 7.643 17.498 32.488 37.307 43.592 46.037 69.849 73.703 87.534 89.174 94.134 112.296 129.857 130.016 134.176 137.470 139.298 165.974 170.242 177.368 189.318 208.388 211.926 241.567 242.312 249.096 254.766 281.712 297.247 311.747 331.115 335.720 338.169 347.561 349.061 359.457 363.064 392.766 410.168 423.859 432.360 455.055 461.081 466.560 478.706 481.930 498.336 503.378 516.944 556.668 563.962 572.032 583.937 606.507 608.571 617.199 625.486 626.171

0.000 7.261 7.375 16.889 31.352 36.009 42.108 44.385 67.394 71.197 84.498 86.066 90.796 108.495 125.544 125.671 129.506 132.850 134.689 160.416 164.492 171.216 183.189 201.776 204.606 232.848 233.531 240.709 244.795 271.460 286.438 299.839 319.702 323.243 325.064 333.930 336.157 346.095 349.714 377.470 394.171 408.458 416.050 437.295 443.572 448.465 460.467 463.403 479.947 483.585 497.031 534.699 543.660 548.612 561.722 581.113 585.293 591.844 600.271 601.676

Table 4.8: 13/16 Carpet Unnormalized Eigenvalues

33

13/16 Alternate Carpet Level: 1 Refinement: 0 n 1 0.000 1.000 2 3 1.023 2.322 4 5 4.427 5.040 6 7 5.885 8 6.283 9.684 9 10 10.299 12.077 11 12 12.481 13.518 13 14 16.020 19.440 15 16 20.692 20.703 17 18 21.837 19 22.168 26.049 20 21 26.851 22 27.620 30.014 23 24 33.939 25 34.358 26 34.663 27 34.956 28 36.408 29 38.808 30 42.401 31 43.803 32 47.542 33 50.717 34 51.509 35 52.784 36 53.046 37 55.189 38 58.168 39 58.559 40 60.645 41 63.797 42 66.750 43 71.942 44 72.679 45 73.850 46 76.182 47 76.523 48 80.073 49 81.101 50 82.967 51 85.302 92.524 52 53 94.789 54 98.859 55 99.688 56 102.700 57 103.748 58 104.432 59 106.486 60 107.349

Normalized Eigenvalues 1 1 2 1 2 0 0.000 1.000 1.022 2.334 4.385 5.033 5.851 6.263 9.481 9.952 11.876 12.198 13.178 15.423 18.651 20.018 20.020 21.012 21.289 24.857 25.342 25.970 27.514 31.482 31.539 32.457 32.568 33.790 35.551 39.369 40.736 41.868 46.180 46.635 47.075 47.781 49.635 51.455 51.503 52.879 56.137 58.157 62.444 64.165 64.510 64.858 66.331 67.831 69.165 70.727 72.674 78.175 78.382 83.804 84.404 84.798 86.146 86.508 87.780 89.333

0.000 1.000 1.021 2.340 4.373 5.038 5.850 6.268 9.436 9.849 11.841 12.139 13.107 15.278 18.467 19.874 19.874 20.827 21.090 24.578 24.974 25.574 26.843 30.792 30.844 31.932 32.016 33.147 34.751 38.610 39.934 40.443 45.034 45.444 45.631 46.529 48.145 49.747 49.755 51.089 54.236 56.034 59.918 61.384 62.256 62.359 63.805 64.644 66.427 67.466 69.526 73.842 74.636 78.865 80.614 80.713 81.812 81.888 82.863 84.523

0.000 1.000 1.018 2.320 4.344 4.966 5.804 6.150 9.354 9.951 11.706 11.967 12.721 15.178 17.715 17.756 18.102 18.760 19.018 22.638 23.240 24.167 26.089 28.737 29.245 32.703 32.704 34.215 34.664 39.376 39.855 42.158 45.742 45.893 46.510 47.897 48.805 49.577 49.589 53.877 55.229 58.707 60.144 63.179 64.117 65.088 65.502 66.082 69.317 69.582 71.732 77.732 79.378 81.191 81.997 84.922 85.365 86.142 88.138 88.235

2 1

2 2

3 0

0.000 1.000 1.018 2.326 4.329 4.965 5.801 6.132 9.309 9.847 11.661 11.891 12.577 15.001 17.396 17.421 17.911 18.413 18.660 22.222 22.798 23.738 25.423 27.980 28.460 32.262 32.330 33.388 34.068 37.931 39.573 41.625 44.451 44.937 45.359 46.625 46.996 48.192 48.529 52.579 54.615 57.057 57.939 61.143 61.949 62.746 64.025 64.489 66.913 67.477 69.390 74.826 75.964 77.383 78.570 81.716 81.840 82.995 84.309 84.404

0.000 1.000 1.017 2.329 4.325 4.966 5.803 6.128 9.298 9.811 11.652 11.870 12.531 14.948 17.286 17.307 17.861 18.299 18.543 22.094 22.662 23.610 25.201 27.740 28.211 32.156 32.255 33.158 33.913 37.500 39.568 41.498 44.076 44.689 45.015 46.266 46.465 47.849 48.329 52.283 54.600 56.422 57.554 60.575 61.377 62.106 63.723 64.152 66.336 67.007 68.813 74.101 75.072 76.146 77.731 80.735 81.010 82.159 83.262 83.353

0.000 1.000 1.016 2.326 4.318 4.960 5.799 6.113 9.282 9.806 11.638 11.854 12.505 14.943 17.291 17.309 17.837 18.297 18.551 22.094 22.655 23.581 25.230 27.790 28.180 32.070 32.164 33.153 33.715 37.388 39.451 41.297 44.032 44.520 44.771 45.992 46.299 47.667 48.166 51.989 54.289 56.256 57.302 60.228 61.093 61.767 63.420 63.824 66.103 66.604 68.456 73.644 74.878 75.560 77.365 80.036 80.612 81.514 82.675 82.868

Table 4.9: 13/16 Carpet Normalized Eigenvalues

34

Figure 4.6:

13 16

Carpet Weyl Ratios, Level 3, 0 Refinements, α = .87537

35

5

Miniaturization

In order to make the ideas clear, we begin by explaining the method of miniaturization on the unit interval I. Here we have a two element group of symmetries consisting of the identity and the reflection ρ(x) = 1 − x about the midpoint. Every Neumann eigenfunction is of the form cos πkx. When k is even, the function is even under ρ, namely u ◦ ρ = u, while if k is odd then the function is odd, namely u ◦ ρ = −u. In this way all eigenspaces are sorted corresponding to the two irreducible representations of the symmetry group. For every even eigenfunction u (except the constant), we can miniaturize it by defining u+ to be ( u ◦ F0−1 on F0 I (5.1) u+ (x) = u ◦ F1−1 on F1 I Note that u◦F0−1 ( 12 ) = u◦F1−1 ( 12 ) because u is even, and the derivative vanishes at 12 because u is a Neumann eigenfunction. This shows that u+ is also a Neumann eigenfunction, and indeed u+ (x) = cos 2πkx. On the other hand, if u is an odd eigenfunction, then define u− by ( u ◦ F0−1 on F0 I u+ (x) = (5.2) −u ◦ F1−1 on F1 I Again u ◦ F0−1 ( 21 ) = −u ◦ F1−1 ( 12 ) because u is odd, so u− is also a Neumann eigenfunction, and again u− (x) = cos 2πkx. We call u+ or u− the miniaturization of u. Note that the representation type of the miniaturization is always even. The eigenvalue of u+ or u− is always 4 times the eigenvalue of u. Thus R = 4 is an eigenvalue renormalization factor. (Of course I has other eigenvalue renormalization factors, namely any square integer, but such luxuries do not generalize to other fractals). Now consider a self-similar fractal with a finite group of symmetries G, and suppose the Laplacian is G invariant. Then each eigenspace splits according to the irreducible representations of G. We seek to find a set of recipes, analogous to (5.1) and (5.2), to miniaturize eigenfunctions according to the corresponding irreducible representations of G. In fact, our goal is to obtain recipes that make sense on the fractal and also on the outer approximating domains. In the latter case the miniaturization of an eigenfunction on Ωm will be an eigenfunction on Ωm+1 . It is by no means clear that this goal is always attainable. We will show explicitly that it is possible for SC, the 12 16 carpet, and the octagasket. In the first two examples the symmetry group is D4 (the dihedral symmetry group of the square), and in the last example it is D8 . In contrast to the interval, the representation type of the miniaturized eigenfunctions is the same as the original one. The referee has pointed out that it is also possible to explain miniaturization on carpets using local reflection maps introduced in [Barlow and Bass 1989] and [Barlow and Bass 1999] (see also Definition 2.12 in [Barlow et al. 2008]). 36

We mention in passing that a version of miniaturization is valid for SG, but the recipes are more complicated. In particular, the multiplicities increase. This is part of the story of spectral decimation (see [Strichartz 2006] for a description). On the other hand, it is not clear how to extend the recipes for the approximating domains Ωm with a positive overlap, although they are presumably valid in the zero overlap case. The symmetry group D4 has five irreducible representations. Let ρH and ρV denote the reflections about the horizontal and vertical axes in D4 , and ρ0D and ρ00D denote the two diagonal reflections. The four one-dimensional representations 1 + +, 1 + −, 1 − +, and 1 − − are characterized by parity with respect to these reflections. (Strictly speaking, we describe functions that transform according to the representations, rather than the abstract representations, since we are interested in eigenfunctions that transform according to representations). Functions transforming according to 1 + + are even with respect to all reflections, and those transforming according to 1 − − are odd with respect to all reflections. The 1 + − functions are odd with respect to ρH and ρV and even with respect to ρ0D and ρ00D , while for 1 − + the reverse holds. Now suppose u is a Neumann eigenfunction on Ωm of 1 + + or 1 − + type. Define the miniaturization u+ = {u ◦ Fi−1 on Fi Ωm } on Ωm+1

(5.3)

for either the SC or 12 16 carpet. On the other hand, for an eigenfunction of 1 + − or 1 − − type define u− = {±u ◦ Fi−1 on Fi Ωm } on Ωm+1

(5.4)

where we alternate the choice of ± on neighboring cells (see Figure5.1). Because of the even or odd parity of u with respect to the reflections ρH and ρV , the miniaturized functions are continuous along the boundaries of the cells of order one. Since u satisfies Neumann boundary conditions, it follows that u+ or u− satisfy matching conditions along these boundaries, hence they are Neumann eigenfunctions on um+1 , and the eigenvalue is multiplied by λ−2 where λ denotes the contraction ratio of the Fi mappings (so λ = 13 for SC and λ = 14 for the 12 12 16 carpet). Note that on the 16 carpet, the miniaturized eigenfunction has the same representation type as u, while on SC, u+ preserves representation type while u− maps 1 + − to 1 + + and 1 − − to 1 − +. There is also a two-dimensional representation of D4 , that we denote by 2. The representation space is spanned by functions u and v satisfying v = ρH u = −ρV u and ρ00D u = −ρ0D u = u, ρ0D v = −ρ00D v = v. The miniaturized functions u2 and v2 are shown in Figure 5.2. Once again we see that u2 and v2 are Neumann eigenfunctions on Ωm+1 with eigenvalue multiplied by λ−2 , and the pair transform according to the representation 2. What does this tell us about the Neumann spectrum on the corresponding fractal? If we believe (1.4) then there will be an eigenvalue renormalization factor R = rλ−2 . For every eigenvalue λn , there will be an eigenvalue equal

37

u -u u -u -u u -u u

u -u u -u u -u u -u

(a)

-u u -u u

(b)

Figure 5.1: 1-D Miniaturized Carpet Eigenfunctions

u -v u v v u -v u

u -v u v u v -u v

-v -u -v -u

v -u v u u v -u v

v -u -v u v u -v -u

-u -v -u -v

(a)

(b)

Figure 5.2: 1-D Miniaturized Carpet Eigenfunctions

38

____?? ____?? ?? ?? ? ____ ? ?? ____??? ? ?_ ___ ??____ ? ? ?? ?? ?? ?____ ____?? ___?? _____ ?? ?? ? ____? ____?? ?? ? ?? ?? ?____ ____ ? ? ? ? ? ?? ____ ?____

____?? ____?? ?? ?? ? ____ ? ?? ____??? ? ?_ ___ ??____ ? ? ?? ?? ?? ?____ ____?? ___?? _____ ?? ?? ? ____? ____?? ?? ? ?? ?? ?____ ____ ? ? ? ? ? ?? ____ ?____

(a)

(b)

u

-v

u

-v

v

-u

-u

v

v

-u

v

-u

u

-v

-v

u

Figure 5.3: The miniaturizations (a) u2 and (b) v2 for a 21 or 23 eigenspace to Rλn with equal multiplicity, and the corresponding eigenfunctions will be miniaturizations as illustrated. But in fact we can run the same miniaturization argument directly on the fractal. Indeed, in both cases we know that there exists a Laplacian ∆ on the fractal satisfying a self-similar identity ∆(u ◦ Fi ) = R−1 (∆u) ◦ Fi

(5.5)

for a certain constant R. Then the miniaturization recipes given above create eigenfunctions with eigenvalue multiplied by R. This is true independent of the validity of the outer approximation method. Incidentally, the miniaturization recipes given above extend easily to any D4 symmetric carpet type fractal. In our last example, the octagasket, the symmetry group is D8 . Here we have four one-dimensional representations. Since D4 ⊂ D8 we may sort the reflections in D8 into those that are in D4 and those that are not. The representation 1++ is described by functions even with respect to all reflections, and 1 − − by all functions odd with respect to all reflections. Similarly, 1 + − functions are odd with respect to D4 reflections and even with respect to all other reflections, while for 1 − + functions the situation is reversed. The miniaturizations u+ (for 1 + + or 1+− eigenfunctions) and u− (for 1−+ or 1−− eigenfunctions) are again given by (5.3) and (5.4), where the ± signs alternate along the eight small octagons. We note that the representation type is preserved under miniaturization. In this case there are three two-dimensional representations, denoted 21 , 22 , 23 . In terms of complex valued functions on the circle, 21 is spanned by e±2πiθ/8 , 22 is spanned by e±2πi2θ/8 , and 23 is spanned by e±2π3iθ/8 . If x, y, z denote any

39

____?? ____?? ?? ?? ? ____ ? ?? ____??? ? ?_ ___ ??____ ? ? ?? ?? ?? ?____ ____?? ___?? _____ ?? ?? ? ____? ____?? ?? ? ?? ?? ?____ ____ ? ? ? ? ? ?? ____ ?____

____?? ____?? ?? ?? ? ____ ? ?? ____??? ? ?_ ___ ??____ ? ? ?? ?? ?? ?____ ____?? ___?? _____ ?? ?? ? ____? ____?? ?? ? ?? ?? ?____ ____ ? ? ? ?? ?____ ??____

(a)

(b)

u

u

-u

-u

-u

-u

u

u

v

-v

v

-v

-v

v

v

-v

Figure 5.4: The miniaturizations (a) u02 and (b) v20 for a 22 eigenspace consecutive points on an eight element orbit of D8 , then 21 functions satisfy √ 2 u(y) = (u(x) + u(z)), (5.6) 2 22 functions satisfy u(x) + u(z) = 0,

(5.7)

and 23 functions satisfy √ u(y) = −

2 (u(x) + u(z)). 2

(5.8)

The 21 and 23 representations have the property that restricted to D4 they become the 2 representation. So if u, v are the basis described above, the miniaturization u2 , v2 are given in Figure 5.3. On the other hand, the restriction of 22 to D4 splits into a direct sum of a 1 + − and a 1 − + representation. So we can choose a basis u, v such that ρH u = ρV u = u = −ρ0D u = −ρ00D u and −ρH v = −ρV v = v = ρ0D v = ρ00D v, and the miniaturization u02 , v20 is given in Figure 5.4. Again the representation type is preserved under miniaturization. Some types of miniaturization on the pentagasket are described in [Adams et al. 2003].

40

6

Random Carpets

For j ∈ Z, j > 1, we partition the unit square into a grid of j by j smaller, equally sized squares of width 1/j. We then randomly remove k of these smaller squares, where k is a small positive integer, and the result is our level 1 domain Ω1 . To produce Ω2 , we partition each square of width 1/j into a grid of j by j equally sized squares of width 1/j 2 , and we then randomly remove m squares of width 1/j 2 from each square of width 1/j. Iterating this process yields a sequence of nested compact domains {Ωm }∞ m=1 where Ωm is a union of squares of side length j −m . Matlab’s rand(’state’) function, a modified version of Marsaglia’s Subtract-with-Borrow algorithm, makes our random choices. The number generator’s state is set according to the exact date and time of the computation, so that the generator’s own state is essentially randomly determined. Also, to shorten FEM computation time we triangulate Ωm with the four sides and two diagonals of each square of side length j −m . The problem we find with our FEM eigenvalue problem on these domains is connectivity. How can we guarantee that each Ωm has only one path component? Also, if two squares are disjoint except at a common vertex, with no other squares in a neighborhood of that vertex, how can we avoid the problem we saw in Section 3? Recall that in this case, the spline space of our finite element solver couples these squares at the common vertex. For simplicity we resolve both questions by choosing small k and altering the above algorithm so that this coupling problem is avoided, as follows. When we pass from Ωm to Ωm+1 we partition a square of side length j −m into squares of side length j −m−1 and delete k of the smaller squares randomly. We then check if this deletion process has produced the above coupling problem. If it has, then we go back and try again; otherwise, we move on to the next mth level square, and so on. For k small enough, the algorithm terminates. Figure 6.1 shows a typical result of the above algorithm. Notice that we have only one path component. Now, we study our spectral information with the eigenvalue counting function N : [0, ∞) → Z, where N (x) is the number of nonnegative eigenvalues less than or equal to x. Then, we examine the Weyl ratio W (x) =

N (x) xα

(6.1)

where xα is an approximate asymptotic bound for N (x), i.e. we choose α ∈ R so that N (x) ∼ xα in accordance with the experimental data. So, finding α corresponds to finding the slope of a linear approximation of N (x) on a loglog plot. In fact, since we are dealing with domains in the plane, the Weyl asymptotic law implies that α = 1 is the correct value as x → ∞. The point is that we truncate our computations well before we reach the region where this asymptotic behavior is approximated, so we observe values of α considerably smaller than 1. In our first example, we let j = 4 and k = 2 and run our algorithm up to level 4 to get {Ωi }4i=1 , where Ω4 is the upper left carpet in Figure 6.2. From this initial carpet, we can restart our algorithm three separate times, beginning 41

Figure 6.1: Level 4 Domain Ω4 for j = 4, k = 3

42

at Ωi once for each i = 1, 2, 3. We then end the algorithm again at level 4 and we call the resulting (level 4) carpet which was started at Ωi the bifurcation of Ω4 at level i + 1. The carpets are shown in Figure 6.2 and the eigenvalue data in Tables 6.1 and 6.2. Next, we let j = 4 and k = 3 and do the same bifurcation study. The carpets are shown in Figure 6.4 and the eigenvalue data in Tables 6.3 and 6.4. Finally, we fix j = 4 and vary k on different levels so that at level 1 we set k = 2, at level 2 we set k = 3, etc. A similar procedure for gaskets rather than carpets is discussed in [Drenning and Strichartz 2008]. Our sequence of k values for the carpet in Figure 6.6 is k = {2, 3, 2, 3, 2}. The eigenvalue data appears in Tables 6.5 and 6.6. The level-to-level eigenvalue ratios in Table 6.5 appear to roughly alternate between the same ratios in Tables 6.3 and 6.1. This is the strongest evidence that the geometry of the domain at different scales is reflected in the spectrum of the Laplacian. Such a correlation is more striking in [Drenning and Strichartz 2008], but the fractals there have a more coherent structure. The Weyl ratios of our first example (where j = 4 and k = 2) appear in Figure 6.3. We now look closely at the agreement of the graph of the original carpet to each individual bifurcation. We see that the original agrees with the bifurcation at Level 4 up to about x = 300, the original agrees with that at Level 3 up to around x = 65, and it agrees with the Level 2 bifurcation up to about x = 25. In our second example (where j = 4 and k = 4) we find the Weyl ratios in Figure 6.5. We do the same comparison. The original agrees with the the Level 4 bifurcation to around x = 150, it agrees with the Level 3 one up to approximately x = 30, and it agrees with the Level 2 bifurcation to approximately x = 10. In other words, the added detail at finer resolutions has only a minimal effect on some initial segment of the spectrum. This is consistent with results in [Drenning and Strichartz 2008]. Our final example’s Weyl ratios (where j = 4 and k = {2, 3, 2, 3, 2}) are found in Figure 6.7. For further comparison of the Weyl ratios, we show those from another trial with j = 4 and k = 2, and those from another trial where j = 4 and k = 3. The carpets for the new j = 4, k = 2 trial appear in Figure 6.8 with Weyl ratios in Figure 6.9, while the carpets for the new j = 4, k = 3 trial appear in Figure 6.10 with Weyl ratios in Figure 6.11. It is clear that different random choices in the construction make a big difference in the spectrum. We leave to the future the problem of formulating precise conjectures concerning the spectra of different random carpets. Acknowledgments: We are grateful to Stacie Goff who contributed to the numerical experiments.

43

Original Carpet

Bifurcation at Level 4

Bifurcation at Level 3

Bifurcation at Level 2

Figure 6.2: Carpet Bifurcations Ω4 for j=4, k=2

44

Original Carpet Level: Refinement: n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

1 2

2 1

3 0

4 0

Bifurc. at Level 4 4 0

5.580 7.666 18.031 31.079 40.933 46.442 49.757 72.354 88.309 96.790 103.355 106.680 138.947 162.163 162.163 168.598 170.390 183.444 198.436 206.000 214.522 240.719 250.543 257.351 266.899 276.814 279.112 302.034 331.424 339.761 372.054 385.306 393.257 395.854 405.837 420.811 424.847 451.376 462.812 502.632 528.577 545.770 551.007 552.842 564.216 578.850 598.075 613.847 631.941 715.621 725.020 727.913 731.705 735.932 736.330 737.370 763.993 770.748 771.833 772.741

4.524 6.734 15.575 24.699 35.373 37.463 41.840 62.389 74.938 77.384 80.835 96.799 114.330 123.498 124.726 138.139 144.519 155.084 172.548 179.213 188.011 196.395 206.038 218.075 224.684 231.084 253.950 265.227 277.522 284.208 299.978 316.612 324.027 339.584 361.323 364.392 375.125 386.611 405.467 408.831 447.618 452.183 464.111 477.614 494.116 501.300 523.765 551.828 558.302 567.952 592.871 600.091 616.830 631.051 637.768 656.446 671.049 681.736 688.547 699.743

3.961 5.914 13.671 21.630 31.097 32.776 36.519 54.211 65.259 65.694 70.391 83.310 90.751 107.533 109.598 117.904 125.422 131.723 152.591 156.432 162.499 170.702 173.663 188.898 195.801 200.575 209.948 223.157 232.146 242.094 251.492 265.554 278.512 294.971 303.434 315.821 320.413 330.029 338.324 347.163 366.670 388.507 394.833 405.344 413.127 431.951 440.593 449.590 477.459 488.904 499.609 503.672 508.602 524.965 532.002 548.081 563.190 573.285 591.230 601.577

3.331 5.008 11.556 18.234 26.152 27.549 30.975 45.450 54.360 55.376 59.301 70.018 74.898 91.169 92.432 99.056 104.987 111.310 128.765 132.112 137.582 143.091 146.420 159.001 164.282 168.720 176.419 187.434 194.488 204.379 212.246 223.592 233.507 248.618 254.390 263.606 268.424 278.419 282.917 292.032 305.564 323.312 329.113 338.488 345.354 362.010 367.808 374.306 399.598 409.795 416.054 419.499 421.296 431.091 438.897 455.651 467.554 472.499 495.228 508.440

3.349 5.009 11.543 18.269 25.983 27.640 30.850 45.767 54.880 55.582 59.323 69.713 75.720 90.934 91.856 99.110 105.280 111.070 129.400 132.276 137.272 143.624 146.806 159.718 164.779 168.320 174.688 187.788 193.716 203.672 211.069 224.233 232.679 248.375 253.273 264.886 268.127 277.801 283.186 290.267 307.685 324.962 330.967 338.553 345.961 361.376 366.624 377.290 402.493 410.350 418.770 422.080 423.380 444.056 447.894 457.588 468.467 493.796 497.748 502.614

Bifurcation at Level 3 3 4 0 0

Bifurcation at Level 2 2 1

3 0

4 0

3.885 5.963 13.514 21.733 31.230 32.326 36.427 54.977 65.444 66.288 70.224 84.547 93.994 105.799 109.995 118.118 128.532 133.795 150.123 158.088 164.643 169.629 173.727 186.572 195.558 201.320 220.595 233.403 238.930 246.169 257.519 266.844 278.277 286.407 301.936 313.507 317.266 325.821 339.347 361.005 387.191 388.234 389.063 405.026 416.434 432.872 444.307 449.181 476.826 480.033 488.846 493.842 505.115 522.488 538.642 555.001 567.954 576.961 581.500 596.141

5.011 6.384 15.597 27.549 35.484 38.315 45.424 56.607 70.649 74.244 83.948 91.564 100.291 118.054 133.591 140.919 149.625 159.175 164.504 177.161 183.438 201.824 209.485 218.375 230.264 235.294 241.413 256.896 278.109 292.874 317.191 331.605 342.980 355.614 371.636 380.539 386.150 397.166 409.025 429.651 448.759 456.622 462.266 486.916 495.591 500.567 512.076 547.600 554.463 576.594 587.455 609.850 612.367 631.828 640.834 648.905 668.772 688.591 705.282 712.415

4.393 5.528 13.664 24.099 31.262 33.722 39.840 49.171 61.489 63.358 72.797 78.693 88.203 102.261 114.904 121.513 134.221 141.125 143.437 154.685 158.717 173.072 180.972 189.522 195.063 204.404 207.898 223.343 245.174 257.954 271.234 287.096 301.702 311.708 324.228 327.137 333.468 346.083 359.088 366.719 389.395 395.283 399.880 414.255 429.603 436.894 438.755 457.011 465.816 477.564 492.776 512.419 521.155 538.115 543.298 549.244 568.342 576.277 604.405 609.218

3.689 4.639 11.478 20.091 26.392 28.249 33.521 41.245 51.576 53.123 60.697 65.923 73.751 85.535 95.879 101.298 112.025 117.324 119.346 129.358 132.738 144.509 151.168 158.937 162.503 171.042 173.358 186.535 203.467 215.964 226.783 238.995 254.300 261.206 269.291 273.775 277.348 287.709 301.704 306.863 325.524 329.216 333.236 346.218 356.793 364.233 367.034 378.966 386.917 397.180 410.999 423.682 428.834 449.802 455.736 458.336 472.118 480.247 502.457 509.436

3.248 4.990 11.311 18.259 26.123 27.255 30.614 45.924 54.860 54.919 59.343 70.628 78.280 88.143 92.539 99.034 108.234 111.329 125.782 133.030 138.022 141.827 146.243 154.557 163.810 170.096 183.338 193.925 199.095 205.403 215.248 222.676 231.720 241.061 253.741 260.755 261.999 270.028 282.063 302.953 316.739 323.430 324.844 335.237 344.196 359.480 363.054 369.326 396.391 397.220 401.513 415.384 419.245 436.796 448.956 459.264 471.301 480.894 488.252 494.604

Original Carpet j Ratios λj+1 n /λn

Table 6.1: Carpet Bifurcation Unnormalized Eigenvalues for j = 4, k = 2

45

0.811 0.878 0.864 0.795 0.864 0.807 0.841 0.862 0.849 0.799 0.782 0.907 0.823 0.762 0.769 0.819 0.848 0.845 0.870 0.870 0.876 0.816 0.822 0.847 0.842 0.835 0.910 0.878 0.837 0.836 0.806 0.822 0.824 0.858 0.890 0.866 0.883 0.857 0.876 0.813 0.847 0.829 0.842 0.864 0.876 0.866 0.876 0.899 0.883 0.794 0.818 0.824 0.843 0.857 0.866 0.890 0.878 0.885 0.892 0.906

0.875 0.878 0.878 0.876 0.879 0.875 0.873 0.869 0.871 0.849 0.871 0.861 0.794 0.871 0.879 0.854 0.868 0.849 0.884 0.873 0.864 0.869 0.843 0.866 0.871 0.868 0.827 0.841 0.836 0.852 0.838 0.839 0.860 0.869 0.840 0.867 0.854 0.854 0.834 0.849 0.819 0.859 0.851 0.849 0.836 0.862 0.841 0.815 0.855 0.861 0.843 0.839 0.825 0.832 0.834 0.835 0.839 0.841 0.859 0.860

0.841 0.847 0.845 0.843 0.841 0.841 0.848 0.838 0.833 0.843 0.842 0.840 0.825 0.848 0.843 0.840 0.837 0.845 0.844 0.845 0.847 0.838 0.843 0.842 0.839 0.841 0.840 0.840 0.838 0.844 0.844 0.842 0.838 0.843 0.838 0.835 0.838 0.844 0.836 0.841 0.833 0.832 0.834 0.835 0.836 0.838 0.835 0.833 0.837 0.838 0.833 0.833 0.828 0.821 0.825 0.831 0.830 0.824 0.838 0.845

Original Carpet Level: Refinement: n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

1 2

2 1

3 0

4 0

Bifurc. at Level 4 4 0

1.000 1.374 3.231 5.570 7.336 8.323 8.917 12.967 15.826 17.346 18.522 19.118 24.901 29.062 29.062 30.215 30.536 32.875 35.562 36.918 38.445 43.140 44.900 46.120 47.831 49.608 50.020 54.128 59.395 60.889 66.676 69.051 70.476 70.942 72.731 75.414 76.138 80.892 82.941 90.078 94.727 97.808 98.747 99.076 101.114 103.737 107.182 110.008 113.251 128.248 129.932 130.451 131.130 131.888 131.959 132.145 136.916 138.127 138.322 138.484

1.000 1.489 3.443 5.460 7.819 8.281 9.249 13.791 16.565 17.106 17.868 21.397 25.272 27.299 27.570 30.535 31.946 34.281 38.141 39.615 41.559 43.413 45.544 48.205 49.666 51.081 56.135 58.628 61.346 62.824 66.310 69.986 71.626 75.064 79.870 80.548 82.921 85.460 89.628 90.371 98.945 99.954 102.591 105.576 109.223 110.811 115.777 121.981 123.412 125.545 131.053 132.649 136.349 139.493 140.977 145.106 148.334 150.696 152.202 154.677

1.000 1.493 3.452 5.461 7.852 8.276 9.221 13.688 16.477 16.587 17.773 21.035 22.914 27.151 27.673 29.770 31.668 33.259 38.528 39.498 41.029 43.101 43.848 47.695 49.438 50.643 53.010 56.345 58.615 61.126 63.499 67.050 70.322 74.477 76.614 79.742 80.901 83.329 85.424 87.655 92.581 98.094 99.691 102.345 104.311 109.063 111.246 113.517 120.554 123.443 126.147 127.172 128.417 132.549 134.325 138.385 142.200 144.749 149.280 151.893

1.000 1.503 3.469 5.473 7.850 8.270 9.298 13.643 16.318 16.623 17.801 21.018 22.483 27.367 27.746 29.734 31.515 33.413 38.652 39.657 41.299 42.952 43.952 47.728 49.313 50.646 52.957 56.263 58.381 61.350 63.711 67.117 70.093 74.629 76.362 79.128 80.574 83.574 84.925 87.661 91.723 97.050 98.792 101.606 103.667 108.667 110.407 112.357 119.949 123.010 124.889 125.923 126.463 129.403 131.746 136.775 140.348 141.833 148.655 152.621

1.000 1.496 3.447 5.455 7.758 8.253 9.211 13.665 16.387 16.596 17.713 20.816 22.609 27.152 27.427 29.593 31.435 33.164 38.637 39.496 40.988 42.885 43.835 47.690 49.201 50.259 52.160 56.071 57.841 60.814 63.023 66.954 69.475 74.162 75.624 79.092 80.060 82.948 84.556 86.670 91.871 97.030 98.823 101.088 103.300 107.903 109.470 112.654 120.180 122.526 125.040 126.028 126.416 132.590 133.736 136.630 139.879 147.442 148.622 150.075

Bifurcation at Level 3 3 4 0 0

Bifurcation at Level 2 2 1

3 0

4 0

1.000 1.535 3.479 5.595 8.039 8.322 9.377 14.152 16.847 17.064 18.077 21.764 24.196 27.235 28.315 30.406 33.087 34.442 38.645 40.696 42.383 43.666 44.721 48.028 50.341 51.825 56.786 60.083 61.506 63.370 66.291 68.692 71.635 73.728 77.725 80.704 81.672 83.874 87.356 92.931 99.672 99.941 100.154 104.263 107.200 111.431 114.375 115.630 122.746 123.572 125.840 127.126 130.029 134.501 138.659 142.870 146.205 148.523 149.692 153.461

1.000 1.274 3.113 5.498 7.082 7.647 9.066 11.297 14.100 14.817 16.754 18.274 20.016 23.561 26.662 28.124 29.862 31.768 32.831 35.357 36.610 40.279 41.808 43.583 45.955 46.959 48.180 51.271 55.504 58.451 63.304 66.181 68.451 70.972 74.170 75.947 77.067 79.265 81.632 85.748 89.562 91.131 92.258 97.177 98.908 99.902 102.198 109.288 110.658 115.075 117.242 121.712 122.214 126.098 127.896 129.506 133.471 137.427 140.758 142.182

1.000 1.258 3.110 5.485 7.116 7.676 9.068 11.192 13.996 14.421 16.570 17.912 20.076 23.276 26.154 27.658 30.551 32.122 32.648 35.208 36.126 39.394 41.192 43.138 44.399 46.525 47.321 50.836 55.805 58.714 61.737 65.347 68.672 70.949 73.799 74.461 75.902 78.773 81.734 83.471 88.632 89.972 91.019 94.290 97.784 99.444 99.867 104.022 106.026 108.701 112.163 116.634 118.622 122.483 123.663 125.016 129.363 131.169 137.571 138.667

1.000 1.257 3.111 5.445 7.153 7.657 9.086 11.179 13.979 14.399 16.451 17.868 19.989 23.183 25.987 27.456 30.363 31.800 32.348 35.061 35.977 39.168 40.973 43.078 44.045 46.359 46.987 50.559 55.148 58.535 61.467 64.777 68.926 70.797 72.989 74.204 75.173 77.981 81.774 83.172 88.230 89.231 90.320 93.839 96.705 98.722 99.481 102.715 104.870 107.652 111.397 114.835 116.231 121.915 123.523 124.228 127.963 130.166 136.186 138.078

1.000 1.536 3.482 5.621 8.042 8.391 9.425 14.138 16.889 16.907 18.269 21.743 24.099 27.135 28.489 30.488 33.320 34.273 38.723 40.954 42.491 43.662 45.022 47.581 50.430 52.365 56.442 59.701 61.293 63.235 66.265 68.552 71.337 74.212 78.116 80.275 80.658 83.130 86.835 93.266 97.510 99.570 100.005 103.205 105.963 110.668 111.768 113.699 122.031 122.287 123.608 127.878 129.067 134.470 138.214 141.387 145.093 148.046 150.311 152.267

Table 6.2: Carpet Bifurcation Normalized Eigenvalues for j = 4, k = 2

46

Original Carpet, α = .84032

Bifurcation at Level 4, α = .83853

Bifurcation at Level 3, α = .85007

Bifurcation at Level 2, α = .83383

Figure 6.3: Weyl Ratios for j = 4, k = 2

47

Original Carpet

Bifurcation at Level 4

Bifurcation at Level 3

Bifurcation at Level 2

Figure 6.4: Carpet Bifurcations Ω4 for j=4, k=3

48

Original Carpet Level: Refinement: n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

1 2

2 1

3 0

4 0

Bifurc. at Level 4 4 0

7.092 11.728 24.546 30.185 42.518 58.544 61.533 77.637 83.257 104.768 113.834 150.330 162.163 162.163 172.077 175.214 188.670 195.969 202.684 240.609 244.457 274.309 280.954 310.772 315.838 325.052 331.424 360.177 385.417 389.957 404.260 415.828 438.883 446.211 459.422 496.933 497.815 508.950 527.137 551.265 567.597 583.940 597.055 604.989 666.885 696.767 724.838 725.020 731.705 733.652 740.114 742.828 760.150 770.961 772.604 809.840 815.252 840.638 884.506 925.356

4.504 8.197 15.759 18.800 27.342 39.332 48.008 55.316 65.486 73.075 80.749 107.208 114.553 123.236 134.430 142.039 151.413 155.785 172.478 181.954 187.959 195.733 213.619 221.652 234.301 249.165 256.361 265.361 283.819 302.005 313.891 319.907 339.911 351.282 358.047 372.237 399.070 411.214 420.241 426.786 450.642 472.465 484.208 488.128 501.510 528.048 544.995 560.630 564.489 579.186 612.913 630.679 646.953 655.027 686.275 696.798 711.641 722.733 746.735 769.117

3.375 6.565 12.132 15.634 22.736 31.633 39.592 44.954 50.292 55.186 63.592 87.607 93.828 97.285 108.788 112.803 120.069 124.187 133.879 145.005 146.514 161.327 164.799 171.028 178.988 191.045 197.974 201.315 227.801 238.307 253.256 261.191 275.572 280.260 294.617 303.439 310.002 318.971 330.335 333.384 355.913 361.077 368.120 377.891 396.592 410.599 416.860 430.081 442.746 445.729 452.752 475.019 497.418 523.642 532.901 544.755 553.161 572.093 586.715 598.816

2.426 4.765 9.007 11.511 16.597 23.474 29.343 33.046 37.106 39.936 47.756 64.387 68.725 71.388 80.883 85.010 88.113 92.302 97.392 105.777 109.927 116.845 120.048 125.428 132.647 141.451 145.517 149.385 162.546 174.063 181.953 192.491 197.205 208.609 217.058 221.374 227.464 233.374 243.651 249.500 265.783 267.320 272.725 278.169 291.488 301.706 305.621 311.371 321.252 328.599 331.363 341.589 364.433 372.127 385.695 393.847 399.410 408.348 413.175 436.475

2.445 4.809 8.969 11.431 17.156 23.415 28.460 33.484 37.063 40.286 46.867 64.396 68.441 72.414 81.781 83.188 88.222 93.366 98.402 106.864 109.130 117.350 120.767 126.050 130.963 139.979 144.028 146.794 167.689 174.823 186.426 192.163 202.084 207.833 218.615 219.487 231.842 233.036 240.021 243.044 258.013 261.429 269.302 275.595 284.793 303.345 305.132 318.362 325.115 329.796 332.623 343.604 360.065 371.947 377.962 396.853 403.815 412.322 425.816 434.588

Bifurcation at Level 3 3 4 0 0

Bifurcation at Level 2 2 1

3 0

4 0

3.493 6.560 12.351 15.081 21.355 29.450 37.253 43.576 50.448 56.936 61.053 83.453 87.807 96.939 105.258 110.001 116.098 125.759 139.613 147.803 153.475 158.670 168.550 176.933 181.365 191.300 202.974 211.209 217.654 234.133 234.618 254.854 263.163 268.417 289.697 294.416 305.155 312.163 323.612 332.791 345.979 352.508 356.974 379.520 390.723 402.895 411.603 418.850 437.407 442.425 460.158 468.450 479.135 500.853 513.718 517.891 544.312 549.102 566.350 576.353

6.127 9.482 21.222 23.081 35.771 42.024 51.343 62.867 69.055 82.013 89.184 104.399 121.120 128.037 141.405 148.384 157.156 166.793 175.989 181.217 189.592 215.292 218.276 240.206 264.516 271.605 277.958 285.512 309.537 315.867 322.848 340.518 357.545 360.353 375.481 409.773 420.361 430.050 442.270 446.720 457.895 464.320 480.125 519.333 534.943 542.283 559.085 571.554 593.800 619.347 624.425 653.206 658.561 663.004 689.445 717.226 723.980 751.200 771.766 801.163

4.897 7.315 16.397 17.654 28.705 34.860 38.677 47.085 53.226 62.297 71.478 78.918 97.702 102.533 111.031 120.540 125.459 129.710 137.547 144.039 153.756 165.384 174.245 186.285 204.123 208.368 218.970 224.683 242.878 251.600 265.116 267.937 271.540 289.120 300.539 314.193 322.296 326.045 336.285 343.015 357.701 367.546 380.877 390.951 417.099 422.342 432.436 444.148 457.423 472.066 485.492 500.482 506.439 520.723 521.847 545.963 575.366 582.018 595.182 615.474

3.695 5.436 12.210 13.228 21.604 25.822 28.823 35.453 39.577 45.968 53.499 58.162 73.003 76.207 82.675 89.074 92.636 96.150 102.029 106.966 113.071 119.406 129.845 134.818 154.817 156.500 162.376 167.545 180.433 187.139 192.433 197.979 198.556 213.750 223.974 228.308 239.171 240.535 246.673 251.129 264.114 271.991 280.646 285.829 303.615 309.099 320.626 325.198 335.551 348.351 356.737 372.797 378.011 380.865 388.528 404.472 428.062 429.608 434.700 448.107

2.631 4.906 9.397 11.247 15.651 21.692 26.997 33.038 38.068 42.856 45.682 61.747 65.440 73.436 78.237 79.855 84.337 91.453 103.979 108.886 113.767 118.341 127.045 132.774 135.580 140.370 148.117 159.046 162.927 173.773 175.749 184.953 194.662 200.150 212.752 216.619 225.425 229.676 237.958 248.954 254.543 260.712 266.488 279.242 287.972 293.498 304.224 306.840 323.790 324.682 336.138 344.312 353.972 366.851 377.732 381.072 395.388 400.724 409.882 421.201

Original Carpet j Ratios λj+1 n /λn

Table 6.3: Carpet Bifurcation Unnormalized Eigenvalues for j = 4, k = 3

49

0.635 0.699 0.642 0.623 0.643 0.672 0.780 0.712 0.787 0.697 0.709 0.713 0.706 0.760 0.781 0.811 0.803 0.795 0.851 0.756 0.769 0.714 0.760 0.713 0.742 0.767 0.774 0.737 0.736 0.774 0.776 0.769 0.774 0.787 0.779 0.749 0.802 0.808 0.797 0.774 0.794 0.809 0.811 0.807 0.752 0.758 0.752 0.773 0.771 0.789 0.828 0.849 0.851 0.850 0.888 0.860 0.873 0.860 0.844 0.831

0.749 0.801 0.770 0.832 0.832 0.804 0.825 0.813 0.768 0.755 0.788 0.817 0.819 0.789 0.809 0.794 0.793 0.797 0.776 0.797 0.780 0.824 0.771 0.772 0.764 0.767 0.772 0.759 0.803 0.789 0.807 0.816 0.811 0.798 0.823 0.815 0.777 0.776 0.786 0.781 0.790 0.764 0.760 0.774 0.791 0.778 0.765 0.767 0.784 0.770 0.739 0.753 0.769 0.799 0.777 0.782 0.777 0.792 0.786 0.779

0.719 0.726 0.742 0.736 0.730 0.742 0.741 0.735 0.738 0.724 0.751 0.735 0.732 0.734 0.743 0.754 0.734 0.743 0.727 0.729 0.750 0.724 0.728 0.733 0.741 0.740 0.735 0.742 0.714 0.730 0.718 0.737 0.716 0.744 0.737 0.730 0.734 0.732 0.738 0.748 0.747 0.740 0.741 0.736 0.735 0.735 0.733 0.724 0.726 0.737 0.732 0.719 0.733 0.711 0.724 0.723 0.722 0.714 0.704 0.729

Original Carpet Level: Refinement: n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

1 2

2 1

3 0

4 0

Bifurc. at Level 4 4 0

1.000 1.654 3.461 4.256 5.995 8.255 8.676 10.947 11.740 14.773 16.051 21.197 22.866 22.866 24.264 24.706 26.604 27.633 28.580 33.927 34.470 38.679 39.616 43.821 44.535 45.834 46.733 50.787 54.346 54.986 57.003 58.634 61.885 62.918 64.781 70.071 70.195 71.765 74.329 77.732 80.035 82.339 84.188 85.307 94.035 98.248 102.206 102.232 103.175 103.449 104.360 104.743 107.186 108.710 108.942 114.192 114.955 118.535 124.721 130.481

1.000 1.820 3.499 4.174 6.071 8.732 10.659 12.281 14.539 16.224 17.928 23.802 25.433 27.361 29.846 31.535 33.617 34.587 38.294 40.398 41.731 43.457 47.428 49.211 52.020 55.320 56.917 58.916 63.013 67.051 69.690 71.026 75.467 77.992 79.494 82.644 88.602 91.298 93.302 94.755 100.052 104.897 107.504 108.374 111.345 117.237 121.000 124.471 125.328 128.591 136.079 140.023 143.637 145.429 152.367 154.703 157.999 160.461 165.790 170.760

1.000 1.945 3.595 4.633 6.737 9.374 11.732 13.321 14.903 16.353 18.844 25.960 27.804 28.828 32.237 33.427 35.580 36.800 39.672 42.969 43.416 47.805 48.834 50.680 53.039 56.612 58.665 59.655 67.503 70.616 75.046 77.398 81.659 83.048 87.303 89.917 91.862 94.519 97.887 98.790 105.466 106.996 109.084 111.979 117.521 121.671 123.526 127.444 131.197 132.081 134.162 140.760 147.398 155.169 157.912 161.425 163.916 169.526 173.859 177.445

1.000 1.964 3.712 4.744 6.840 9.674 12.093 13.619 15.292 16.459 19.681 26.536 28.324 29.421 33.334 35.035 36.314 38.040 40.138 43.593 45.304 48.155 49.475 51.692 54.667 58.296 59.972 61.566 66.990 71.736 74.988 79.331 81.273 85.974 89.455 91.234 93.744 96.180 100.415 102.826 109.536 110.170 112.397 114.641 120.130 124.341 125.955 128.325 132.397 135.425 136.564 140.778 150.193 153.363 158.955 162.315 164.608 168.291 170.281 179.883

1.000 1.967 3.668 4.675 7.016 9.576 11.640 13.694 15.158 16.476 19.167 26.336 27.991 29.616 33.446 34.022 36.081 38.184 40.244 43.705 44.632 47.993 49.391 51.551 53.561 57.248 58.904 60.035 68.581 71.499 76.244 78.590 82.647 84.999 89.409 89.765 94.818 95.306 98.163 99.399 105.521 106.918 110.138 112.712 116.474 124.061 124.792 130.202 132.964 134.879 136.035 140.526 147.258 152.117 154.578 162.303 165.151 168.630 174.149 177.736

Bifurcation at Level 3 3 4 0 0

Bifurcation at Level 2 2 1

3 0

4 0

1.000 1.878 3.536 4.317 6.113 8.431 10.665 12.475 14.442 16.300 17.478 23.891 25.137 27.752 30.133 31.491 33.236 36.002 39.968 42.313 43.937 45.424 48.252 50.652 51.921 54.765 58.107 60.465 62.310 67.027 67.166 72.959 75.338 76.842 82.934 84.285 87.360 89.366 92.643 95.271 99.047 100.916 102.194 108.649 111.856 115.340 117.833 119.908 125.221 126.657 131.734 134.108 137.166 143.384 147.067 148.262 155.825 157.197 162.134 164.998

1.000 1.548 3.464 3.767 5.839 6.859 8.380 10.262 11.272 13.387 14.557 17.040 19.770 20.899 23.081 24.220 25.652 27.225 28.726 29.579 30.946 35.141 35.628 39.208 43.176 44.333 45.370 46.603 50.524 51.557 52.697 55.581 58.360 58.819 61.288 66.885 68.614 70.195 72.190 72.916 74.740 75.789 78.369 84.768 87.316 88.514 91.257 93.292 96.923 101.093 101.922 106.620 107.494 108.219 112.535 117.069 118.172 122.615 125.972 130.770

1.000 1.494 3.349 3.605 5.862 7.119 7.899 9.616 10.870 12.723 14.597 16.117 19.953 20.940 22.675 24.617 25.622 26.490 28.090 29.416 31.401 33.775 35.585 38.044 41.687 42.554 44.719 45.886 49.602 51.383 54.143 54.719 55.455 59.045 61.377 64.166 65.821 66.586 68.678 70.052 73.051 75.062 77.784 79.842 85.182 86.252 88.314 90.706 93.417 96.407 99.149 102.210 103.427 106.344 106.574 111.499 117.504 118.862 121.551 125.695

1.000 1.471 3.305 3.580 5.847 6.989 7.801 9.595 10.712 12.441 14.479 15.742 19.758 20.625 22.376 24.108 25.072 26.023 27.614 28.950 30.603 32.317 35.143 36.488 41.901 42.357 43.947 45.346 48.834 50.649 52.082 53.583 53.739 57.852 60.619 61.792 64.732 65.101 66.762 67.968 71.483 73.614 75.957 77.360 82.174 83.658 86.778 88.015 90.817 94.281 96.551 100.898 102.309 103.081 105.155 109.471 115.855 116.274 117.652 121.280

1.000 1.865 3.572 4.275 5.950 8.246 10.263 12.559 14.471 16.291 17.366 23.473 24.876 27.916 29.741 30.356 32.060 34.765 39.527 41.392 43.248 44.987 48.295 50.473 51.540 53.361 56.305 60.460 61.936 66.059 66.810 70.309 73.999 76.086 80.876 82.346 85.694 87.310 90.458 94.638 96.763 99.108 101.304 106.152 109.471 111.571 115.648 116.643 123.087 123.426 127.781 130.888 134.560 139.456 143.592 144.862 150.304 152.332 155.814 160.117

Table 6.4: Carpet Bifurcation Normalized Eigenvalues for j = 4, k = 3

50

Original Carpet, α = .80788

Bifurcation at Level 4, α = .80253

Bifurcation at Level 3, α = .82004

Bifurcation at Level 2, α = .81408

Figure 6.5: Weyl Ratios for j = 4, k = 3

51

Figure 6.6: Level 4 Domain Ω4 for j = 4, D:2,3,2,3,2

52

Level: Refinement: n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Eigenvalue Data 1 2 2 1 7.812 11.846 16.892 31.783 38.849 44.579 66.179 79.132 91.235 93.671 115.299 118.662 162.163 162.163 163.646 170.709 174.715 188.946 201.351 204.687 244.350 247.462 264.337 274.203 281.799 290.260 325.832 331.424 336.355 361.265 384.370 394.115 409.812 411.970 439.508 440.834 460.863 498.602 510.810 523.667 531.231 546.687 559.577 577.182 581.677 598.344 626.841 633.797 703.078 717.798 725.020 731.705 737.699 741.151 745.273 765.924 769.461 771.967 784.362 819.314

5.846 8.843 11.188 21.564 27.674 33.739 53.267 57.301 64.613 71.615 72.824 89.806 95.052 106.641 118.545 125.049 135.278 150.973 161.952 170.606 182.087 182.669 202.217 210.504 218.830 227.945 234.838 266.942 289.198 308.120 312.120 316.748 335.456 343.218 359.087 363.915 375.298 386.461 405.711 413.699 430.848 437.950 440.856 471.699 479.787 498.649 504.051 518.365 527.983 547.394 563.709 579.899 592.267 605.924 622.510 648.652 671.205 672.947 697.036 722.128

3 0

4 0

5 0

5.041 7.739 9.621 18.685 24.037 28.815 45.651 50.164 55.715 60.579 65.016 79.457 83.209 90.290 98.672 102.625 113.855 120.531 138.235 144.171 153.859 155.701 175.672 181.596 186.415 196.993 202.526 224.635 236.851 250.806 257.020 264.292 274.591 289.720 304.720 310.972 321.912 336.603 342.497 345.709 352.572 369.987 377.907 383.805 394.602 409.685 421.465 430.774 439.453 449.694 453.597 471.292 491.538 508.950 512.829 543.566 547.092 553.461 579.292 598.492

3.855 5.918 7.327 14.339 18.351 21.870 34.817 38.328 42.305 45.932 48.923 60.983 63.326 63.836 74.716 78.056 86.167 91.075 105.178 108.697 110.986 115.922 132.411 137.161 141.338 150.327 153.692 172.469 179.569 187.350 194.251 200.424 205.929 216.514 232.617 233.960 241.177 245.761 255.459 264.311 265.834 281.332 286.941 291.534 298.709 313.466 321.144 326.844 333.442 334.683 344.470 358.560 371.267 382.992 387.189 410.317 413.608 418.171 434.757 448.196

3.160 4.865 6.019 11.813 15.056 17.953 28.614 31.538 34.727 37.773 40.398 50.215 51.851 52.366 61.516 63.851 70.686 74.673 86.257 89.387 90.735 95.112 108.679 112.420 115.948 123.167 126.141 141.217 147.123 153.399 160.344 164.390 168.413 177.282 190.896 191.294 196.161 201.687 209.461 217.396 218.537 230.437 235.441 239.439 245.557 255.821 263.599 268.188 272.346 273.914 283.262 294.274 303.822 314.146 317.207 336.800 339.283 343.727 356.559 367.600

j Ratios λj+1 n /λn j=1 j=2 j=3

j=4

0.748 0.746 0.662 0.678 0.712 0.757 0.805 0.724 0.708 0.765 0.632 0.757 0.586 0.658 0.724 0.733 0.774 0.799 0.804 0.833 0.745 0.738 0.765 0.768 0.777 0.785 0.721 0.805 0.860 0.853 0.812 0.804 0.819 0.833 0.817 0.826 0.814 0.775 0.794 0.790 0.811 0.801 0.788 0.817 0.825 0.833 0.804 0.818 0.751 0.763 0.778 0.793 0.803 0.818 0.835 0.847 0.872 0.872 0.889 0.881

0.820 0.822 0.821 0.824 0.820 0.821 0.822 0.823 0.821 0.822 0.826 0.823 0.819 0.820 0.823 0.818 0.820 0.820 0.820 0.822 0.818 0.820 0.821 0.820 0.820 0.819 0.821 0.819 0.819 0.819 0.825 0.820 0.818 0.819 0.821 0.818 0.813 0.821 0.820 0.822 0.822 0.819 0.821 0.821 0.822 0.816 0.821 0.821 0.817 0.818 0.822 0.821 0.818 0.820 0.819 0.821 0.820 0.822 0.820 0.820

0.862 0.875 0.860 0.867 0.869 0.854 0.857 0.875 0.862 0.846 0.893 0.885 0.875 0.847 0.832 0.821 0.842 0.798 0.854 0.845 0.845 0.852 0.869 0.863 0.852 0.864 0.862 0.842 0.819 0.814 0.823 0.834 0.819 0.844 0.849 0.855 0.858 0.871 0.844 0.836 0.818 0.845 0.857 0.814 0.822 0.822 0.836 0.831 0.832 0.822 0.805 0.813 0.830 0.840 0.824 0.838 0.815 0.822 0.831 0.829

0.765 0.765 0.762 0.767 0.763 0.759 0.763 0.764 0.759 0.758 0.752 0.768 0.761 0.707 0.757 0.761 0.757 0.756 0.761 0.754 0.721 0.745 0.754 0.755 0.758 0.763 0.759 0.768 0.758 0.747 0.756 0.758 0.750 0.747 0.763 0.752 0.749 0.730 0.746 0.765 0.754 0.760 0.759 0.760 0.757 0.765 0.762 0.759 0.759 0.744 0.759 0.761 0.755 0.753 0.755 0.755 0.756 0.756 0.750 0.749

Table 6.5: Carpet Mixed Unnormalized Eigenvalues and Ratios, 23232

53

blah blah Level: Refinement: n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

1 2

2 1

3 0

4 0

5 0

1.000 1.516 2.162 4.068 4.973 5.706 8.471 10.129 11.679 11.991 14.759 15.190 20.758 20.758 20.948 21.852 22.365 24.186 25.774 26.201 31.278 31.677 33.837 35.100 36.072 37.155 41.709 42.425 43.056 46.244 49.202 50.449 52.459 52.735 56.260 56.430 58.994 63.824 65.387 67.033 68.001 69.980 71.630 73.883 74.459 76.592 80.240 81.130 89.999 91.883 92.807 93.663 94.431 94.872 95.400 98.044 98.496 98.817 100.404 104.878

1.000 1.513 1.914 3.689 4.734 5.771 9.112 9.802 11.052 12.250 12.457 15.362 16.259 18.242 20.278 21.390 23.140 25.825 27.703 29.183 31.147 31.247 34.591 36.008 37.432 38.991 40.171 45.662 49.469 52.706 53.390 54.182 57.382 58.710 61.424 62.250 64.197 66.107 69.399 70.766 73.699 74.914 75.411 80.687 82.071 85.297 86.221 88.670 90.315 93.635 96.426 99.195 101.311 103.647 106.484 110.956 114.814 115.112 119.232 123.525

1.000 1.535 1.908 3.707 4.768 5.716 9.056 9.951 11.052 12.017 12.897 15.762 16.506 17.911 19.574 20.358 22.585 23.910 27.422 28.599 30.521 30.887 34.848 36.023 36.979 39.078 40.175 44.561 46.984 49.752 50.985 52.428 54.471 57.472 60.447 61.688 63.858 66.772 67.941 68.578 69.940 73.394 74.966 76.136 78.277 81.269 83.606 85.453 87.174 89.206 89.980 93.490 97.507 100.961 101.730 107.827 108.527 109.790 114.914 118.723

1.000 1.535 1.901 3.720 4.761 5.673 9.032 9.943 10.975 11.916 12.692 15.820 16.428 16.561 19.383 20.249 22.354 23.627 27.286 28.198 28.792 30.073 34.351 35.583 36.666 38.998 39.871 44.743 46.584 48.603 50.393 51.995 53.423 56.169 60.346 60.695 62.567 63.756 66.272 68.569 68.964 72.984 74.439 75.631 77.492 81.320 83.312 84.791 86.503 86.825 89.364 93.019 96.315 99.357 100.446 106.446 107.300 108.483 112.786 116.273

1.000 1.540 1.905 3.739 4.765 5.681 9.055 9.981 10.990 11.954 12.785 15.892 16.409 16.572 19.468 20.207 22.370 23.632 27.298 28.288 28.715 30.100 34.394 35.577 36.694 38.979 39.920 44.691 46.560 48.546 50.744 52.024 53.298 56.104 60.413 60.539 62.079 63.828 66.288 68.799 69.160 72.926 74.510 75.775 77.711 80.960 83.421 84.873 86.189 86.686 89.644 93.129 96.150 99.418 100.387 106.587 107.373 108.779 112.840 116.334

Table 6.6: Carpet Mixed Normalized Eigenvalues 23232

54

Figure 6.7: Weyl Ratios for j = 4, k = {2, 3, 2, 3, 2}, Level 5 Carpet, α = .8071

55

Original Carpet

Bifurcation at Level 4

Bifurcation at Level 3

Bifurcation at Level 2

Figure 6.8: Carpet Bifurcations Ω4 for j=4, k=2

56

Original Carpet, α = .84747

Bifurcation at Level 4, α = .84753

Bifurcation at Level 3, α = .83368

Bifurcation at Level 2, α = .83019

Figure 6.9: Weyl Ratios for j = 4, k = 2

57

Original Carpet

Bifurcation at Level 4

Bifurcation at Level 3

Bifurcation at Level 2

Figure 6.10: Carpet Bifurcations Ω4 for j=4, k=3

58

Original Carpet, α = .81013

Bifurcation at Level 4, α = .80544

Bifurcation at Level 3, α = .82086

Bifurcation at Level 2, α = .81975

Figure 6.11: Weyl Ratios for j = 4, k = 3

59

References [Adams et al. 2003] Bryant Adams, S. Alex Smith, Robert S. Strichartz, and Alexander Teplyaev, The spectrum of the Laplacian on the pentagasket, Fractals in Graz 2001, Trends Math., Birkh¨auser, Basel, 2003, pp. 1-24. MR MR2091699 (2006g:28017) [Barlow 1995] Martin T. Barlow, Diffusions on fractals, Lecturs on probability theory and statistics (Saint-Flour, 1995), Lecture Notes in Math., vol. 1690, Springer, Berlin, 1998, 1-121. MR MR1668115 (2000a:60148) [Barlow and Bass 1989] M. T. Barlow, R. F. Bass, The construction of Brownian motion on the Sierpinski carpet, Ann. Inst. H. Poincar´e Probab. Statist. 25 (1989), no. 3, 225-257. MR MR1023950 (91d:60183) [Barlow and Bass 1999] M. T. Barlow, R. F. Bass, Brownian motion and harmonic analysis on Sierpinski carpets, Canad. J. Math. 51 (1999), no. 4, 673-744. MR MR1701339 (2000i:60083) [Barlow et al. 1990] M. T. Barlow, R. F. Bass, and J. D. Sherwood, Resistance and spectral dimension of Sierpinski carpets, J. Phys. A 23 (1990), no. 6, L253-L258. MR MR1048762 (91a:28007) [Barlow et al. 2008] M. T. Barlow, R. F. Bass, T. Kumagai, and A. Teplyaev, Uniqueness of Brownian motion on Sierpinski carpets, preprint. [Blasiak et al. 2008] Anna Blasiak, Robert S. Strichartz, and Baris Evren Ugurcan, Spectra of Self-Similar Laplacians on the Sierpinski Gasket with Twists, Fractals 16 (2008), no. 1, 43-68. MR MR2399872 (2009c:35331) [Broman and Camia 2008] Erik I. Broman and Federico Camia, Large-N limit of crossing probabilities, discontinuity, and asymptotic behavior of threshold values in Mandelbrot’s fractal percolation process, Electron. J. Probab. 13 (2008), no. 33, 980-999. MR MR2413292 [Chayes et al. 1988] J. T. Chayes, L. Chayes, R. Durrett. Connectivity properties of Mandelbrot’s percolation process, Probab. Theory Related Fields 77 (1988), no. 3, 307-324. MR MR0931500 (89d:60193) [Colin de Verdi`ere 1998] Yves Colin de Verdi`ere, Spectres de graphes, Cours Sp´ecialis´es [Specialized Courses], vol. 4, Soci´et´e Math´ematique de France, Paris, 1998. MR MR1652692 (99k:05108) [Dalrymple et al. 1999] Kyalle Dalrymple, Robert S. Strichartz, and Jade P. Vinson, Fractal differential equations on the Sierpinski Gasket, J. Fourier Anal. Appl. 5 (1999), no. 2-3, 203-284. MR MR1683211 (2000k:31016) [Dodziuk and Patodi 1976] J. Dodziuk and V. K. Patodi, Riemannian structures and triangulations of manifolds, J. Indian Math. Soc. (N.S.) 40 (1976), no. 1-4, 1-52 (1977). MR MR0488179 (58 #7742) 60

[Drenning and Strichartz 2008] Shawn Drenning and Robert Strichartz, Spectral Decimation on Hambly’s Homogeneous Hierarchical Gaskets, (to appear) MR [Kigami 2001] Jun Kigami, Analysis on fractals, Cambridge Tracts in Mathematics, vol. 143, Cambridge University Press, Cambridge 2001. MR MR1840042 (2002c:28015) [Kuchment and Zeng 2001] Peter Kuchment and Hongbiao Zeng, Convergence of Spectra of Mesoscopic Systems Collapsing onto a Graph, J. Math. Anal. Appl. 258 (2001), 671-700. MR MR1835566 (2002d:35158) [Strichartz 1967] Robert S. Strichartz, Multipliers on fractional Sobolev spaces, J. Math. Mech. 16 (1967), 1031-1060. MR MR0215084 (35 #5927) [Strichartz 1999] Robert S. Strichartz, Analysis on fractals, Notices Amer. Math. Soc. 46 (1999), no. 10, 1199-1208. MR MR1715511 (2000i:58035) [Strichartz 2003] Robert S. Strichartz, Fractafolds based on the Sierpinski gasket and their spectra, Trans. Amer. Math. Soc. 355 (2003), no. 10, 4019-4043 (electronic). MR MR1990573 (2004b:28013) [Strichartz 2006] Robert S. Strichartz, Differential Equations on Fractals: A Tutorial, Princeton University Press, Princeton, NJ 2006. MR MR2246975 (2007f:35003) Department of Mathematical Sciences, George Mason University, Fairfax, VA 22030 E-mail address: [email protected] Department of Mathematics, Cornell University, Ithaca, NY 14850-4201 E-mail address: [email protected] Department of Mathematics, Cornell University, Ithaca, NY 14850-4201 E-mail address: [email protected]

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