Gaps in the spectrum of the Laplacian on $3N$-Gaskets

This article develops analysis on fractal $3N$-gaskets, a class of post-critically finite fractals which include the Sierpinski triangle for $N=1$, specifically properties of the Laplacian $\Delta$ on these gaskets. We first prove the existence of a self-similar geodesic metric on these gaskets, and prove heat kernel estimates for this Laplacian with respect to the geodesic metric. We also compute the elements of the method of spectral decimation, a technique used to determine the spectrum of post-critically finite fractals. Spectral decimation on these gaskets arises from more complicated dynamics than in previous examples, i.e. the functions involved are rational rather than polynomial. Due to the nature of these dynamics, we are able to show that there are gaps in the spectrum.


Definitions and main results
For a fixed natural number N > 0, a 3N -Gasket is a post-critically finite self-similar fractal K with a set of 3N maps F i : K → K, i = 1, 2, . . . , 3N . K has fractal boundary V 0 = {v 1 , v 2 , v 3 }, and the maps F i are determined by the rule F i (v 2 ) = F j (v 3 ) where i = j + 1 mod 3N and F kN (v 1 ) = v k for k = 1, 2, 3.
This defines a self-similar structure in the sense of [Kig93,Kig01] by the theory of ancestors in [Kig93]. The limiting self-similar structure is K is defined as the quotient of the shift space Σ = {1, 2, . . . , 3N } N of infinite words on the alphabet of length 3N with quotient map π : Σ → K. Here v 1 = π(N ) where N := N N N · · · is the infinite string of N 's, and F i (v 1 ) = π iN . Thus π is defined by the equivalence relation π w(i)(2N )N = π w(j)(3N )N where w ∈ {1, 2, . . . , 3N } k is a finite word and i ≡ j + 1 mod 3N .
In the case where N is not divisible by 4, these fractals are the attractors of the iterated function system consisting of the 3N maps G i : R 2 → R 2 , G i (x) = r(x − p i ) + p i , where p i are the corners of a regular 3N -gon, r = sin(π/3N ) sin(π/3N ) + sin(π/3N + 2πm/3N ) , and m = 3N/4 . In the case when N = 1, the fractal is the Sierpiński gasket. In the case where N is divisible by 4, the above definition defines an infinitely ramified fractal, and does not fit into the framework of the current work. We approximate K by the sets V n ⊂ K, n = 0, 1, 2, . . ., where and V 0 is the fractal boundary.
We think of V n as the vertex set of a graph, where x, y ∈ V n are adjacent if there is w = i 1 i 2 · · · i n ∈ {1, 2, . . . , 3N } n , such that x and y ∈ F w (V 0 ). In this case we write x ∼ y.
For a set V , let (V ) = {f : V → R}. If V is finite (V ) = R |V | . We define the Laplacian on a given 3N -gasket as the scaled limit of graph Laplacian on the sets V n . Specifically, we define the operator ∆ n : (V n ) → (V n ) by where d x is the degree of x in the graph above. For f ∈ (K), and x ∈ V k for some k, define We define the domain of ∆ to be the functions f for which the above limit exists and is finite for all x ∈ V * and, moreover, represents a continuous function on V * . Then, in this case, ∆f (x) can be continued from V * to a continuous function on the fractal. In [Kig01, BCF + 07] it is shown that the limit is non trivial and has a unique extension to a self-adjoint operator, which we also denote ∆, with discrete spectrum σ(∆).
Since (V n ) is a finite dimensional vector space for all n, ∆ n can be thought of as a matrix, and the spectrum σ(∆ n ) is the eigenvalues of this matrix. For example, for any N , and thus σ(∆ 0 ) = {0, 3/2}. Take d n to be the graph distances on the nth level approximating graphs, that is We prove in Section 3 that when scaled properly, the sequence {d n } ∞ n=1 converges to d * which can be extended to a self-similar geodesic distance on the whole of the 3N -gasket K and induces the standard topology.
The heat kernel p(t, x, y) : R + × K × K → R of the operator ∆ with respect to µ is the integral kernel such that, if f ∈ L 2 (K) and u : then u satisfies the differential equation With respect to the distance d * and the measure µ, we get the following heat kernel estimate Theorem 2.1. The Laplacian ∆ on K has a symmetric heat kernel p(t, x, y) : R + ×K ×K → R which satisfies the following estimates Here d H = log(3N )/ log(N +1) is the Hausdorff dimension with respect to d * , d w = α+d H = (log(2N + 3) + log(N ))/ log(N + 1) where α = log ρ/ log(N + 1), ρ = c/(3N ) = 1 + 2N/3, and c 1 ,c 2 ,c 3 ,c 4 are positive real constants.
The proof of this theorem is given in Section 3. We note here that the symmetry of the heat kernel is a result of the fact that ∆ is self-adjoint with respect to the self-similar measure µ. This follows from [Kig01].
Remark 2.2. Note that the spectral dimension was first computed in [BCF + 07] in different notation. Our current ρ was denoted by c in [BCF + 07], and our current N was denoted by 3N in [BCF + 07]. According to the terminology of [BCF + 07], our 3N gasket is a (N, N, N )-gasket, which means N 1 = N 2 = N 3 = N (but in [BCF + 07] the notation was N = N 1 + N 2 + N 3 , which would contradict our use of N ).
We use the technique called spectral decimation to calculate the spectrum of ∆ n+1 from the spectrum of ∆ n . Thinking of ∆ n as a matrix, the first step in spectral decimation is computing the rational functions R(z) and φ(z) such that where the matrix-valued function S(z) = (A − z) + B(D − z) −1 C is the Schur compliment of with block representation such that A corresponds to the vertices in the nth level approximation.
Theorem 2.3. For the 3N -gasket, N ≥ 1, R(z) and φ(z) are the rational functions if N is odd, if N is odd. (2.5) Further, for N > 1, R(z) has simple poles at the set of points cos 2 mπ 2N : m = 1, 3, . . . , N − 1 if N is even, and (2.6) For N > 1, the set of points above along with the point 3/2 are the complete set of zeros of φ(z). This is proved in the appropriate subsections of Section 4. Above T k and U k are Chebyshev polynomials of the first and second type respectively, that is to say that T k (z) = cos(k arccos(z)) and U k (z) = sin((k + 1) arccos(z)) sin(arccos(z)) for the appropriate domains, and are extended by the polynomial representation everywhere else.
An interesting feature of this result is that the function R(z) is given by a rational function which has disconnected real Julia set (see [HSTZ12]) because R(z) has poles in the convex hull of its Julia set. One implication of this is that the spectrum of the renormalized limit of ∆ n , ∆ has gaps in the sense of [Str05], which implies that the Fourier series of continuous functions converge uniformly on K.
Knowing the functions R(z) and φ(z) allow us to calculate the spectrum of ∆ n as follows.
Theorem 2.5. If ∆ n is the graph Laplacian of the nth approximating graph to a 3N -Gasket, then ∆ n is a matrix of dimension 3N + (3N − 2)(3N ) n 3N − 1 and the set of eigenvalues is The multiplicities of these eigenvalues is as follows (i) The multiplicity of 0 is 1, and the multiplicity of 3/2 is (ii) For any n ≥ 1 and 0 ≤ m < n − 1 such that R m (z) = sin 2 (kπ/(3N )) ∈ σ(∆ 1 ) k = 1, 2, . . . 3N − 1, then the multiplicity of z is (iii) For any z with R m (z) ∈ A, n ≥ 0, and 0 ≤ m < n − 2, the mulitiplicity of z is The proof of this theorem is given at the end of Subsection 6.2. We defineR to be the branch of the inverse of R(z) such thatR(0) = 0, that isR•R(z) = z for z in a neighborhood of 0. Using the above theorem and equation 2.1, one has Theorem 2.6. If, as before, ∆ = lim n→∞ c n ∆ n is the Laplcian on K, with c = 3N + 2N 2 , then the spectrum This follows from calculations similar that in [Str06, Chapter 3], and is a technique which goes back to [FS92].
We also are able to calculate the normalized limiting distribution of eigenvalues, that is, the limit as n → ∞ of the normalized probability measures κ n defined to be where mult n (z) is the multiplicity of z as an eigenvalues of ∆ n . The following results is implied by Theorem 2.5.
Corollary 2.7. Normalizing with by the number of eigenvalues including multiplicity, the limiting distribution of eigenvalues (the integrated density of states) is a pure point measure κ = lim n→∞ κ n with the set of atoms The value of κ at these atoms is given as follows In the previous results, we assumed that N is fixed, and considered limits as n → ∞. There are two different kind of limits. One corresponds to the graph Laplacians ∆ n , that converge as n → ∞ to a Laplacian on an infinite graph, with limiting density of states κ. The other limit corresponds to the limits of renormalized Laplacian c n ∆ n , that converge to the continuous Laplacian ∆ on our fractal, 3N -gasket. Eigenvalues of ∆ are given in Theorem 2.6. Our results allow one to explicitly compute the limits of these objects as N → ∞. , and its spectral decimation with respect to the circular graph of 3N vertices. This means that we consider the process of spectral decimation that geometrically can be described as removing the teeth from the sawtooth graph. The sawtooth spectral decimation function R sawtooth (z) is computed in lemma 4.1 to be R sawtooth (z) = 2z. The density of states of the probabilistic Laplacian on the circular graph of 3N vertices converges as N → ∞ to an absolutely continuous measure on [0, 2]. By applying R sawtooth (z) to this continuous measure, and using Corollary 2.7, we obtain the result.
(2) The eigenvalues of the circular graph of 3N vertices are and so the eigenvalues of the circular sawtooth graph, using the function R sawtooth (z) as in lemma 4.2, are They have to be multiplied by c = 3N + 2N 2 , which yield the result, taking into account that the function R(z) is approximately linear in the neighborhood of zero.
Remark 2.9. Following arguments in Corollary 2.8, one can show that, as N → ∞, the eigenfunctions converge uniformly to the usual sin and cos eigenfunctions on the circle, as our 3N -fractals K converge, in the Gromov-Hausdorff sense, to the usual circle. See [BCF + 07, Pos12] for some related results.

Metrics, measure, energy and the heat kernel
In this section we prove Theorem 2.1, using techniques from [Kig12,BN98]. At the heart of these techniques, is relating a geodesic metric on the space to the resistance metric. To define the resistance metric, let for f ∈ (V n ) be the graph energy of V n . It is shown in [BCF + 07, Kig01] that It is also established that the Laplacian ∆ defined in Section 2 is the infinitesimal generator of E as a Dirichlet form on L 2 (K, µ). For more on Dirichlet forms, see [FOT11,CF12].
We define the effective resistance between two points x, y ∈ K as follows It is also established [Kig01] that R(x, y) is a metric on K.
The reader should not confuse this resistance metric R(x, y) with the rational function R(z) heavily studied in our paper but not in the current section. Notation for R(x, y) and R(z) is well-established in the literature, and we do not to wish to change the tradition, even though it may be confusing within one paper.
We now define the geodesic metric on our fractal spaces.
Proposition 3.1. There exists a metric d * on K, inducing the original topology, with the following properties On the other hand, if w ∈ {1, 2, . . . , 3N } n , for some constant c ≤ 3 + 1/N . (iv) K is a geodesic space with d * as a metric.
. , x k = y is a length minimizing path, then we can extend this to a path in V n+1 by paths which connect x i to x i+1 of length N + 1. This path can be seen to be the shortest path because . . , 3N } n , thus any path in V n+1 between x and y would induce a path in V n by only considering elements in that path which are also in V n . Thus d n+1 (x, y) = (N + 1)d n (x, y) for all x, y ∈ V n . Thus the metric d * on We shall prove (i-iii) for d * defined on V * , which will imply that d * is jointly continuous with respect to the subspace topology of V * ⊂ K. This in turn implies that d * extends to a metric on K which induces the original topology, and that (i-iii) are satisfied by the extended metric. The fact that K is compact proves that d * is complete.
We have taken (i) as a definition, and (ii) can be seen inductively from the argument above. It is left to prove (iii).
Since the F i are d * -similitudes with scaling factor (N + 1) −1 , it is enough to show that the above is true with n = 0. First, given Thus by the triangle inequality, we have our bound on d * (x, y).
(iv) For points on V * , it is easy to see that d * has approximate midpoints: i.e., for every x, y ∈ V * , and ε > 0, there is z ∈ K such that |d(x, y)/2 − d(y, z)| ≤ ε. In this case we can take z from V k for k large enough. By the standard theory of metric spaces, see [BBI01], this proves that d * is a geodesic metric.
(v) follows because K is the attractor of the iterated function system generated by the 3N F i , which are each similitudes with respect to d * with Lipshitz constant 1/(N + 1).
Proof. Part (i) follows from proposition 7.16 (b) in [BN98] which states there is a constant , and then using s proof similar to that of part (iii) of proposition 3.1. Part (ii): There are at most two w in {1, 2, . . . , 3N } n , such that to be the harmonic extension of the function g which is 1 at F −1 w (x) and 0 at the other boundary points. Noting that E (g) = ρ/2, then There is a constant c such that for all x, y ∈ K and α = log ρ/ log(N + 1).
Proof. Define to be the union of n-cells which contain x or intersect a cell which contains x. If y ∈ N n (x), then by proposition 3.
On the other hand, if y / ∈ N n (x), any path from x to y must cross two elements of V n (possibly including x). We conclude d * (x, y) ≥ c 3 (N + 1) −n . The function defined in the proof of 3.2 vanishes outside of N n (x), so R(x, y) ≥ c 3 ρ −n . T  T  T  T  T        T  T  T  T  T        T  T  T  T  T        T  T  T  T  T        T  T  T  T  T        T  T  T Proof of Theorem 2.1. We shall walk through the requirements of Theorem 15.10 of [Kig12]. The property (ACC) is implied by R(x, y) being uniformly perfect for local Dirichlet forms, see [Kig12, Proposition 7.6]. Our Dirichlet form is local, and R(x, y) is uniformly perfect which is implied by proposition 3.3 and the fact that d is geodesic. The fact that d ∼ QS R is also implied by proposition 3.
It also follows from proposition 3.1 part (iii) that there are constants c 5 and c 6 such that which along with proposition 3.3 proves the (DM2) g,d * . The chain condition is also implied by the fact that d * is geodesic. Thus Theorem 2.1 follows from [Kig12, Theorem 15.10] using condition (b).

Computation of R(z), φ(z) and the exceptional set
Consider the sawtooth graph G 2 with vertex set {v 0 , . . . , v m , u 1 , . . . , u m } and edge relation v i ∼ v i+1 , v i ∼ u i , and v i ∼ u i+1 for i = 0, 1, . . . , m − 1. Let L 2 be the graph Laplacian of G 2 . The sawtooth graph is depicted in Figure 4.1.
We consider the eigenvalue problem with boundary v 0 and v m , that is to say that to find f ∈ (G 2 ) and z ∈ C such that given prescribed values for f (v 0 ) and f (v m ).
Proof. First consider the subgraph G 1 consisting of the vertexes v 0 , v 1 , . . . , v m , and the Laplacian L 1 on this subgraph. It is known that functions of the form g(v k ) = e ikθ , for θ to be determined, have eigenvalues 1 − cos(θ) except, potentially, at the endpoints v 0 and v m . Suppose f is an eigenfunction of L 2 with eigenvalue z 2 , except at v 0 and v m . Then, for k ∈ {1, . . . , m − 1}, so by dividing by 1 − z 2 in (4.3), substituting the result into (4.4), and collecting our f (v k )'s we get By further simplifying (4.5), we get In particular, if z = 3/2, f | G 1 is an eigenfunction of L 1 with eigenvalue z 1 = 2z 2 . But since, for an eigenfunction f , the values at u k are determined by the eigenvalue and the functions value at v k and v k+1 , we can assume z 1 = 1 − cos(θ).
In addition, both sin(kθ) and cos(kθ) are both eigenfunctions themselves on the line graph with eigenvalues of the same form. It is easy to see the f 1 and f 2 above are eigenfunctions with eigenvalue z as prescribed, and that this technique does not work for z outside of that range. Further, since f 1 (v 0 ) = −f 1 (v m ) = 1 and f 2 (v 0 ) = f 2 (v 2 ) = 1, any value at the v 0 and v 2 points can be achieved by a linear combination of these two function.
It is easy to check that the functions {g j } m j=0 are linearly independent and are 3/2eigenfunctions of L 2 (notice that these function are eigenfunctions on the whole of G 2 , not just away from boundary).
To prove that these are all such eigenfunctions, possibly with boundary, we proceed by induction on m. It is clearly true for m = 1. Define the subgraph of where the ± is determined by the parity of m, is zero away from the vertices of G m−1 2 , and thus when restricted is a 3/2-eigenfunction with boundary of G m−1 2 , and thus must be in the span of {f j } m−1 j=0 by our induction hypothesis. Lemma 4.2. For a 3N -gasket, the spectrum of ∆ 1 , the Laplacian of the first level approximating graph is (4.7) σ(∆ 1 ) = 3 2 sin 2 jπ 3N : j = 0, 1, . . . , 3N − 1 .
where the multiplicity of 3/2 is 3N and where the multiplicity of the other eigenvalues is given by their multiplicity in (4.7).
Proof. Using terminology from the proof of lemma 4.1, the first level approximating graph of a 3N -gasket can be thought of as G 2 for m = 3N and the identification of boundary points v 0 = v 3N . It is straight forward to check that f (v k ) = e iθk is still an eigenfunction of G 1 with the aforementioned identification, and that it satisfies the eigenvalue problem at v 0 = v 3N .
The other 3N eigenfunctions all have eigenvalue 3/2. The 3/2-eigenspace is span by, for example g j − cg 1 , for j = 0, 2, 3, . . . , 3N where c ∈ {−1, 0, 1} is chosen so the values at v 0 and v 1 match. It will be important later to note that all but three of these functions have Dirichlet boundary conditions, that is, they take value 0 everywhere on V 0 . This linear sawtooth eigenbasis (4.1) and (4.2) will be incredibly useful in computing our spectral decimation function R(z). As it will simplify some calculations later on, we take as shorthand We wish to find expressions for P and Q (and hence l and r) in terms of our eigenfunctions. Accordingly, (1 + f 1 (v 1 )) in the symmetric case and in the skew-symmetric case. These expressions give us (1 + f 1 (v 1 )) (4.11)

Computation of R(z).
Having an eigenbasis and a set of eigenvalues, we now wish to find the spectral decimation function on the 3N -Gasket, which we will call R(z). Let g be an arbitrary eigenfunction of ∆ n+1 on the sawtooth graph with eigenvalue z, and shall assume a priori that g| Vn has eigenvalue R(z) (which will turn out to be expressed as a rational function of z). As we wish to look at the restriction to V n . We consider two adjacent n-level cells isomorphic to V 1 cells which intersect at a point which we will call x 3N , assign it the eigenfunction the value g(x 3N ) = A at this vertex. Moving counterclockwise, we sequentially call the other two points in V 0 x 4N and x 5N and assign the value B to each. We can assume this symmetry because we will not be checking the the eigenvalue condition at x 4N and x 5N , and thus can replace the values of g there with there averages. We label the vertices in V 1 of degree 4, x 0 , ...x 3N −1 , beginning at the vertex immediately counterclockwise from x 3N and continuing counterclockwise. Similarly, label the degree 2 points x 3N , x 3N +1 , . . . , x 6N −1 . By symmetry we can assume that g is symmetric under exchanging the two n-cells adjacent to x 3N .
To simplify things, we consider a function h ∈ (V 1 ) to locally build g out of, by defining h(x 3N ) = 1 and h(x 4N ) = h(x 5N ) = 0 and which is extended to the rest of V 1 by the relation With this scheme we get (1 + a + ar + bl) Simplifying 4(1 − z)a = 1 + a + ar + bl (4.13) Returning our focus to the four vertices adjacent to x in V n , we see that ∆ n g| Vn (x) = R(z)A = A − 1 4 (4B).On the other hand, taking linear combinations of rotations of h, we get the value aA + (b + c)B at vertices adjacent to x 3N in V n+1 . Thus zA = A − (aA + (b + c)B). Combining this with (1 − R(z))A = B, we get zA = A − A(a + (b + c)(1 − R(z))) which yields (4.14) We now wish to put R(z) entirely in terms of z (suppressing the n + 1 subscript). Thus, we add the three equations in 4.13 and solving for a + b + c, getting To calculate b + c in the denominator, we combine equations (4.13) to get Now, solving these three equations for b + c yields and thus With (4.19), (4.11), (4.12), (4.1) and (4.2) we have all of the ingredients we need to construct our function R(z). With much computer aid we found the above to be equivalent to the form in Theorem 2.3.

Computation of φ(z).
We would like to compare the V 1 approximation to the V 0 approximation, so we treat function g of our Laplacian in vector in block form where g 0 = g| V 0 and g 1 = g| V 1 \V 0 . Consider ∆ 1 as a 6N × 6N matrix with the above block structure written 6N − 3) matrix and B and C are appropriately sized rectangular matrices. If we further assume g is an eigenfunction of ∆ 1 away from the boundary V 0 , then (4.20) Focusing on the second equation given to us by (4.20), we see that is the Schur complement of the matrix ∆ 1 − z, and S 1 1 (z) is the first entry in the first row of this Schur complement. We compute that Now, to find φ(z) we must first find S 1,1 (z). Accordingly, we multiply the vector u 0 = (1, 0, 0) T by S(z), and use (4.21) and the fact that A is a 3 × 3 identity matrix to get The rows of B correspond to boundary points, Therefore, the first entry of Bu 0 is equal to −1/2 times the sum of the value of u 1 at each vertex adjacent to the boundary point with value 1. Since these boundary conditions correspond directly to the eigenbasis we picked for our fractal when calculating R(z), the boundary point in question is only adjacent to two other vertices, both of which have value a, using the notation from Subsection 4.3, Bu 0 = −a.
Substituting this value into our equality for S 1,1 (z) we get S 1,1 (z) = 1 − z − a. Thus, using this fact and (4.14) we have In particular, which we had solved for in terms of l and r earlier. Using a computer algebra system, we obtained the simplified version in Theorem 2.3.

4.3.
Poles of R(z) and zeros of φ(z). We prove that the singularities of R(z) are of the form 2.6. First, note that T N and U n have the N unique roots , the extema of T N (x) must be zeros of U N −1 and the value at these points must be ±1, see, for example [MH03].
Consider the case when N is even. Then it is clear from (2.4) and the previous discussion that the singularities of R(z) must be contained in the N/2 zeros of T N ( √ z), We have where negatives are determined in the last equalities depending on the parity of the degree of the Chebyshev polynomial. But the form of the extrema of T N (x) shows that (4.22) is always equal to ±1 and the form of the roots of U N indicates (4.23) is always zero. Thus, 2 )θ N )) = 0 as the roots of U N −1 (x) are exactly of the form cos( mπ N ). Therefore, the numerator of R(z) in the case where N is even is never zero so no singularity in this case is removable. The proof is similar for N odd.

Gaps in the spectrum
In this section we follow [HSTZ12] to establish properties of the gaps in the spectrum of the Laplacian on a 3N -gasket in the sense of [Str05]. We say that an infinite non-negative increasing sequence {α i } ∞ i=1 has gaps, if lim sup α k+1 /α k > 1. Proposition 5.1. If we take z max to be the max {z : R(z) = z or R(z) = 0} and I 0 = [0, z max ]. If there are is a pole of R in I 0 , then there are gaps in the spectrum of ∆.
Proof. The Julia set of R(z), call it J , is contained in in R −n (I 0 ), for n = 0, 1, 2, . . .. Because there is a pole in I 0 , R −n (I 0 ) R −n+1 (I 0 ), and is a finite set of intervals. Since J = I 0 , it must be completely disconnected.
This implies that for every ε ≥ 0, there is a sub interval of [0, ε] which is not contained in J . Which implies that there are gaps in the spectrum of J by an argument similar to the un-numbered Theorem in Section 3 of [HSTZ12].
Proof of Theorem 2.4. From Theorem 2.3, the poles of R(z) are calculated to be real and between 0 and 1. Also, we see that there are zeros of R(z) at the point, includes 1 when N is even and cos 2 (π/2N ) if N is odd. Either way there is a zero of R(z) greater than the largest pole. Thus by Theorem 5.1 there are gaps in the spectrum.
Written as a block matrix, Proposition 6.1. The exceptional set  Here the dimension of the eigenspace of the eigenvalues 3/2 is 3N − 3, eigenvalues in A is 2, and the multiplicity of the rest is counted by multiplicity of occurrences in the set.
First we classify the eigenvalues of D which are also Eigenvalues of ∆ 1 , which we refer to as Dirichlet-Neumann eigenvalues.
Proof. In the proof of lemma 4.2, we have already shown that 3/2 is in σ(D), and has multiplicity at least 3N − 3. By a similar argument as that employed at the end of the proof of lemma 4.1, it is not difficult to prove that the multiplicity is exactly 3N − 3.
An element in σ(∆ 1 ) is either 3/2 of of the form Thus, either By choosing appropriate j divisible by 3 and between 1 and 3N we can find any m between 1 and N/2. Thus all poles of R(z) are in σ(∆ 1 ). Further, if z 0 ∈ σ(∆ 1 ) and z 0 = 3/2 then we have shown that the corresponding eigenfunction f 0 is an extension of the eigenfunction from the circle.
Lemma 6.4. The set σ(D)\σ(∆ 1 ) consists of the N elements of A, each being an eigenvalue with multiplicity 2, and the N/2 + 1 elements of B, each with multiplicity 1.
Proof. Considering ∆ 1 as a block matrix as before Thus, if z 0 / ∈ σ(∆ 1 ) then lim z→z 0 S −1 (z) is well defined. We prove that for z 0 / ∈ σ(∆ 1 ) the multiplicity of z 0 as an eigenvalue of D is equal to the the dimension of the nullspace lim z→z 0 S −1 (z).
Since S(z 0 ) exists and is invertable for z 0 / ∈ σ(D), the statement is vacuously true there. On the other hand, if z 0 ∈ σ(D) then there is f ∈ (V 0 \ V 1 ) such that And in particular lim z→z 0 S −1 (z)(Bf ) = 0. Thus the matrix B is a linear mapping from the z 0 -eigenspace of D to the nullspace of lim z→z 0 S −1 (z)).
Conversely, assuming lim z→z 0 S −1 (z)g = 0 for some g ∈ (V 0 ) with g = 0, because z 0 / ∈ σ(∆ 1 ) (∆ 1 − z 0 ) −1 is well defined, Thus, by the block representation we have non-zero f such that (D − z)f = 0. Thus the linear map g → lim z→z 0 −(D − z) −1 CS −1 (z)g is a well defined map from the nullspace of lim z→z 0 S −1 (z) to the z 0 -eigenspace of D, and is clearly the inverse of the map above, which proves that the two spaces have the same dimension.
If v = [1, 1, 1] T and u ∈ span v ⊥ , i.e. v is invariant under all symmetries of the 3N -gasket. It is simple to compute .
Now assume z 0 ∈ A then R(z 0 ) = 0 and φ(z) has a pole at z 0 and lim z→z 0 R(z)φ(z) = 0 and is well defined. Thus S −1 (z 0 )u = 0 and S −1 (z)v = 0, and so the nullspace of S −1 (z 0 ) = 2. Thus, z 0 corresponds to an eigenspace of dimension 2, span by anti-symmetric eigenfunctions with respect to reflection about a given axis. A geometric argument shows that there should be N such eigenfunctions for a given axis of symmetry (so 2N linearly independent functions in total).
On the other hand, for z 0 ∈ B, then both φ(z) and φ(z)R(z) have a pole at z 0 . Thus S −1 (z 0 )v = 0, so z 0 corresponds to an eigenfunction which is invariant under all symmetries. A geometric argument shows that there should be (N + 2)/2 such eigenfunctions if N is even and (N + 1)/2 if N is odd.
Adding up the multiplicities we have found all of the 3N − 3 eigenvalues of D.
6.2. Multiplicities. In this section we recall from [BCD + 08a] and prove new results about spectral decimation which will allow us to determine the eigenvalues with multiplicities of ∆ n . Using these formulas we will be able to determine the spectrum of the Laplacian on the limiting fractal. We define mult M (z) to be the multiplicity of an eigenvalue z of the matrix M , and dim M to be the dimension of a square matrix M . Additionally mult n (z) = mult ∆n (z) and dim n = dim ∆n . Using previous work in [BCD + 08a], we have the following proposition, letting m denote the number of contractions defining an iterated function system (i.e. for a 3N -gasket, m = 3N ): (1) If z / ∈ E(M 0 , M ), then (6.4) mult n (z) = mult n−1 (R(z)), and every corresponding eigenfunction at depth n is an extension of an eigenfunction at depth n − 1.
In computing the spectrum of the Laplacian of the 3N gasket, we shall need the following additional case, which is not covered by the result above.
Proof. In Theorem 3.3 of [BCD + 08a] it is shown that (∆ n − z) −1 = (D − z) −1 P n + (P n−1 − (D − z) −1 C)(φ(z)∆ n−1 − φ(z)R(z)) −1 (P n−1 − B(D − z) −1 P n ) Where P n−1 the orthogonal projection on (V n ) to the set of functions supported on V n−1 , and P n = I n − P n−1 is the projection on the orthogonal complement i.e. functions supported away from V n−1 . Using the spectral decomposition ∆ n = λ∈σ(∆n) λP n,λ where P n,λ is the orthogonal projection onto the λ-eigenspace of ∆ n , we get λ∈σ(∆n) 1 λ − z P n,λ = (D − z) −1 P n + λ ∈σ(∆ n−1 ) (P n−1 − (D − z) −1 C) 1 φ(z)(λ − R(z)) P n−1,λ (P n−1 − B(D − z) −1 P n ) Multiplying both sides by (z 0 − z) and taking the limit z → z 0 , the left hand side of the above becomes P n,z 0 (the projection will be zero when z 0 is not an eigenvalue). The first term on the right hand side becomes In other words, if f is an eigenfunction of z 0 , f | V n−1 ≡ 0. We establish in the end of Section 4 that eigenvectors of z 0 for ∆ 1 are such that if u is a z 0 -eigenfunction of D then the values on V 0 of ∆ 1 u are orthogonal to constant vectors, i.e.
Further, for any vector in R 3 orthogonal to constants, there is a z 0 -eigenfunction of D such that ∆ 1 u has that vector as boundary values.
In other words, if we consider the graph G with vertex set F w (V 1 ) | w ∈ {1, 2, . . . , 3N } n−1 are n − 1 cells of the V n , and edge relation is given if intersection is non-empty. A z 0eigenfunction f induces a function J on the edges of G, by where x is the unique element in the intersection of F w (V 1 ) and F v (V 1 ). Further J is such that J(α, β) = −J(β, α) and β∼α J(α, β) = 0 (in this sum, α is fixed). Additionally, any function J with these two properties can be obtained in this way. The dimension of the linear space spanned by such functions is the 1st Betti number of the graph, which is easily computed to be