Vibration modes of 3n-gaskets and other fractals

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1 Vibration modes of n-gaskets and other fractals N Bajorin, T Chen, A Dagan, C Emmons, M Hussein, M Khalil, P Mody, B Steinhurst, A Teplyaev teplyaev@mathuconnedu Department of Mathematics, University of Connecticut, Storrs CT 0669 USA Abstract We study eigenvalues and eigenfunctions (vibration modes) on the class of self-similar symmetric finitely ramified fractals which includes n-gaskets We consider such examples as the Sierpinski gasket, a non-pcf analog of the Sierpinski gasket, the level- Sierpinski gasket, a fractal -tree, the hexagasket, and one dimensional fractals We develop a theoretical matrix analysis, including analysis of singularities, which allows us to compute eigenvalues, eigenfunctions and their multiplicities exactly We support our theoretical analysis by symbolic and numerical computations AMS classification scheme numbers: 8A80, C, B, 60J, 9C99 PACS numbers: 00Sa, 00Bb, 00Ga, 060Lj, 070Hm Introduction In this paper we study eigenvalues and eigenfunctions (vibration modes) on the class of self-similar fully symmetric finitely ramified fractals Such studies originated in [0, ], where it was observed that on the Sierpiński there are highly localized eigenfunctions corresponding to eigenvalues of very high multiplicity Later the spectrum of the Laplacian on the Sierpiński gasket was studied in detail in [], and an example of the modified Koch curve was studied in [, ] The main purpose of our paper is to develop a theoretical matrix analysis, including analysis of singularities, which allows the computation of eigenvalues, eigenfunctions and their multiplicities for a large class of more complicated fractals Our analysis, in particular, allows the computation of the spectral zeta function on fractals (see [8, 9]) and the limiting distribution of eigenvalues (ie integrated density of states) The latter is a pure point measure, except in the examples which are based on the one dimensional interval This support has a representation supp(κ) = J R D, where J R is the Julia set of a rational function, which we compute, and D is a possibly empty set of isolated points (if D is infinite, it accumulates to J R ) Also, our analysis allows the computation of eigenvalues and eigenfunctions by a highly accurate hierarchical iterative procedure, which does not involve large matrix calculations and is illustrated in Figures, and see

2 Vibration modes of n-gaskets and other fractals Figure A basic Neumann eigenfunction on the Sierpiński gasket, three dimensional views There is a large body of physics and mathematics literature devoted to analysis on fractals A small sample of it, containing many references, is [,,, ] and [,,, 6,,, 6, 7, 8, 0] In particular, tools for the numerical analysis of the Sierpiński gasket were developed in [7, 6], and fractal antenae were considered in [,, 7, 9] Our study is closely related to the analysis of self-similar graphs [7, 8, 9,, 6,, and references therein], quantum graphs [0,, and references therein], selfsimilar groups [, 7, 8, 9, 8,, and references therein], and the relation between electrical circuits and Markov chains [6, 0,, and references therein] This paper is organized as follows In Section we give the definition of the finitely ramified fractals with full symmetry, on which the graphs which we consider are based In Section we introduce spectral self-similarity, Schur complement and a Drichlet-to- Neumann map, and show how the resolvent of the Laplacian can be computed by an iterative procedure In Section we analyze the singularities of our map and obtain general formulas for eigenvalues and their multiplicites We also obtain formulas for corresponding eigenprojectors In the subsequent sections we use our general method to analyze the following examples: the Sierpiński gasket (Section ), a non-pcf analog of the Sierpiński gasket (Section 6), the level- Sierpiński gasket (Section 7), a fractal -tree (Section 8), the hexagasket (Section 9), the unit interval as a self-similar set

3 Vibration modes of n-gaskets and other fractals Figure A basic Neumann eigenfunction on the level- Sierpiński gasket, three dimensional views (Section 0), and the diamond fractal (Section ) Finitely ramified fractals with full symmetry A compact connected metric space F is called a finitely ramified self-similar set if there are injective contraction maps ψ,,ψ m : F F such that m F = ψ i (F) i=

4 Vibration modes of n-gaskets and other fractals Figure A Neumann eigenfunction on the level- Sierpiński gasket, three dimensional views and for any n and for any two distinct words w,w W n = {,,m} n we have F w F w = V w V w, where F w = ψ w (F) and V w = ψ w (V 0 ) It is assumed that V 0 is a finite set of at least two points, which often is called the boundary of F Here for a finite word w = w w n W n we denote We define V n = ψ w = ψ w ψ wn m ψ i (V n ) = i= w W n V w and call this set the vertices of level or depth n There is a natural infinite self-similar sequence of fractal finite graphs G n with vertex set V n defined as follows For each n 0 and w W n we define G w as a complete graph with vertices V w Then, by definition, G n = w W n G w Note that G n has no loops, but is allowed to have multiple edges, depending on the structure of the fractal F, as in Section 6 The degree of a vertex x in graph G n is denoted by deg n (x) The degrees of vertices are uniformly bounded in all our examples except the non-pcf analog of the Sierpiński gasket in Section 6

5 Vibration modes of n-gaskets and other fractals The main object of our study are eigenvalues and eigenfunctions on the probabilistic graph Laplacians n on G n, which are defined by n f(x) = f(x) deg n (x) (x,y) E(G n) f(y) where E(G n ) denotes the set of edges of the graph G n For convenience we denote the matrix of n by M n in the standard basis of functions on V n Our main geometric assumption is that for any permutation σ : V 0 V 0 there is an isometry g σ : F F that maps any x V 0 into σ(x) and preserves the self-similar structure of F This means that there is a map g σ : W W such that ψ i g σ = g σ ψ gσ(i) for all i W The group of isometries g σ is denoted by G It is well know that the eigenvalues and eigenfunctions of n describe vibration modes of so called cable systems modeled on the graph G n They are also can be considered as discrete approximations to eigenvalues and eigenfunctions of a continuous self-similar Laplacian µ on F This continuous self-adjoint Laplacian is the generator of a self-similar diffusion process on F which can be defined in the standard way in terms of a self-similar resistance (Dirichlet) form on F, that is for any f in a suitably defined domain Dom µ of the Neumann Laplacian we have E(f,f) = f µ fdµ where µ is the standard suitably normalized self-similar (Hausdorff, Bernoulli) measure on F A G-invariant resistance form E on F is self-similar with energy renormalization factor ρ if for any f Dom(E) we have E(f,f) = ρ F m E(f i,f i ) i= Here we use the notation f w = f ψ w for any w W n Such resistance forms in the case of pcf fractals were studied in detail in [] The finitely ramified case can be studied in a similar way because of the general results in [] In particular, existence and uniqueness, up to a scalar multiplier, of the local regular self-similar G-invariant resistance form E is shown in [0] Moreover, one can show that where the usual graph energy is E n (f,f) = E = lim n ρ n E n (x,y) E(G n) ( f(x) f(y) ) and that (ρm) n n f(x) n µf(x)

6 Vibration modes of n-gaskets and other fractals 6 for any function f for which µ f C(F) and any x V = n 0 V 0 In addition, one has a relation ρm = d dz R(0) > where R(z) is the rational function that appears in the spectral decimation process, and is one of the most important objects in our study The standard and almost trivial example of the self-similar energy and Laplacian in a finitely ramified situation is the case of F = [0,] In this case we can take m = with ψ (x) = x and ψ (x) = x+, the self-similar measure µ is the usual Lebesgue measure, µ f = f and E(f,f) = 0 (f (x)) dx = 0 ff dx = F f µ f dµ for any f Dom( µ ) = {f : f L [0,],f (0) = f () = 0} Then we of course have ρ = and n n f(x) = f(x) f(x ) f(x + n ) n n f (x) n for any f C [0,] The cases F = [0,] with m = and m = are discussed in Section 0 Although in general the fractal F is an abstract metric space, in our examples F R and the metric on F is the restriction of the usual Euclidean metric in R Moreover, the isometries g σ are restrictions of isometries of R that maps F into itself and preserves the self-similar structure of F We do not require that contractions ψ i are similitudes (see Section 6) One can easily construct more involved and higher dimensional examples for which our methods apply Spectral self-similarity, Schur complement and Drichlet-to-Neumann map If we have a matrix M given in a block form [ A B M = C D ] () then its Schur complement is A BD C () In our work one of the most important objects is the Schur complement of the matrix M z which is defined by S(z) = A z B(D z) C () Note that we use a convention that M z denotes M zi where I is the identity matrix of the same size as M Similarly, A z and D z denote the matrices A and D minus z times the identity matrix of the appropriate size Our interest in S(z) can be explained as follows As the initial step in our calculations, we would like to relate the eigenvalues and eigenvectors of the larger Laplacian matrix M = M and the eigenvalues and eigenvectors of a smaller Laplacian

7 Vibration modes of n-gaskets and other fractals 7 matrix M 0 In our setup, the blocks A and D in () correspond to outer (boundary) and interior vertices respectively Suppose v is an eigenvector of M which is partitioned into its boundary part v 0 and interior part v Then eigenvalue equation can be written as [ A B C D or as two equations Mv = zv ] [ v0 v ] [ v0 = z v Av 0 + Bv = zv 0 Cv 0 + Dv = zv From the second equation we obtain v = (D z) Cv 0, provided z / σ(d), which implies S(z)v 0 = 0 (6) If v 0 is also an eigenvector of M 0 with an eigenvalue z 0, then we would like to relate (6) with (M 0 z 0 )v 0 = 0 (7) According to [7, 6], we can write z 0 = R(z) if we solve what is our main equation ( ) S(z) = φ(z) M 0 R(z), (8) where φ(z) and R(z) are scalar (meaning not matrix-valued) rational functions Proposition For a given fully symmetric self-similar structure on a finitely ramified fractal F there is a unique rational function φ(z) and R(z) that solve equation (8) Proof Clearly S(z) is a matrix valued rational function By our main symmetry assumption in the previous section, for any z the matrix S(z) is a linear combination of the identity matrix and M 0, which implies the proposition Remark From the calculations above one can see that S(λ) is the so called Drichlet-to-Neumann map for the Laplacian In our examples M 0 is a matrix that has on the diagonal and diagonal Therefore we have that and φ(z) = (N 0 )S, (z) R(z) = S,(z) φ(z) ] N 0 Here N 0 is the number of boundary vertices, which is the number of points in V 0 From the calculations above we have the following theorem () () off the

8 Vibration modes of n-gaskets and other fractals 8 Theorem Suppose that z is not an eigenvalue of D, and not a zero of φ Then z is an eigenvalue of M with an eigenvector [ ] v if and only if R(z) is an eigenvalue of v0 M 0 with an eigenvector v 0, and v = v where v = (D z) Cv 0 This implies, in particular, that there is an one-to-one map from the eigenspace of M 0 corresponding to R(z) onto the eigenspace of M corresponding to z where v 0 v = T(z)v 0 T(z) = I 0 (D z) C Naturally, the map v 0 v is called the eigenfunction extension map, and T(z) is called the eigenfunction extension matrix The theorem above suggest the following definition of the so called exceptional set E(M 0,M) = σ(d) {z : φ(z) = 0} Once we have computed the functions R(z) and φ(z) using the smaller matrices M 0 and M = M, we can compute the spectrum of much larger matrices M n by induction using the following results We use notation [ ] An B M n = n for the block decomposition of M n corresponding to the representation C n D n where V n = V n \V n V n = V n V n Theorem For all n > 0 we have a relation P n (M n z) P n = φ(z) (M n R(z)), where P n is defined as the restriction operator from V n to V n We often identify P n with the orthogonal projection from l (V n ) onto the subspace of functions with support in V n Suppose that z n / E(M 0,M) Then z n is an eigenvalue of M n with an eigenvector v n if and only if z n = R(z n ) [ ] vn is an eigenvalue of M n with an eigenvector v n, and v n = where v n = (D n z n ) C n v n In such a situation v n is called the continuation of the eigenfunction v n from V n to V n \V n v n

9 Vibration modes of n-gaskets and other fractals 9 One can obtain information about the extension of eigenfunctions and eigenprojectors from V n to V n by the following theorem Theorem Let P n,zn be the eigenprojector of M n corresponding to an eigenvalue z n / E(M 0,M), and P n,zn be the eigenprojector of M n corresponding to eigenvalue z n = R(z n ) Then where P n,zn = φ(z n ) d dz R(z n) T n(z n )P n,zn (P n B n (D n z n ) P n) (9) T n (z) = (P n (D n z) C n ) and P n is defined as the restriction operator from V n to V n \V n We often identify P n with the orthogonal projection from l (V n ) onto the subspace of functions that vanish on V n In this case P n = I n P n Proof First we will prove the key formula for the proof of these theorems This formula is not related to spectral similarity and is a known fact Essentially, it shows how to find the inverse of a matrix given in a two-by-two block form To simplify notation we assume that n = and M = M Suppose that matrices D x and A x B(D x) C are invertible Then M x is invertible and (M x) = (D x) + + (P 0 (D x) C)(A x B(D x) C) (P 0 B(D x) ) (0) It is enough to prove this formula for x = 0, ie to prove M = D + (P 0 D C)(A BD C) (P 0 BD ) () provided that D and A BD C are invertible We have MD = (P + P 0 )MD P = P + P 0 MD P and Thus P 0 M(P 0 D C) = MP 0 P MP 0 P 0 MD C = P 0 (A BD C) M(D P + (P 0 D C)(A BD C) (P 0 BD P )) = = P + P 0 MD P + P 0 (P 0 BD P ) = P + P 0 = I That is what () says To obtain the proof Theorem, note that (0) implies (M x) = (D x) P + + (P 0 (D x) C)(φ(x)M 0 φ (x)) (P 0 B(D x) P ), () where φ (z) = φ(z)r(z) The statements of Theorem follow if we use the standard spectral representation M = zp z z σ(m) and pass to the limit as x z in this formula

10 Vibration modes of n-gaskets and other fractals 0 Remark For n = these theorems are also true for the adjacency matrix graph Laplacian For n > it is important that we consider probabilistic graph Laplacian, or a multiple of it For instance, [9, 6] and related works usually consider the Laplacian, n, multiplied by Analysis of the exceptional values It is not enough to restrict ourself to values of z outside of the exceptional set E(M 0,M) In fact, this set is very interesting because it often contains eigenvalues of high multiplicity, which in turn often correspond to localized eigenfunctions We first formulate a proposition that gives the multiplicities of such eigenvalues, and is used extensively to analyze examples in the rest of the paper Then we prove a theorem which implies the proposition We write mult n (z) for the multiplicity of z as an eigenvalue of M n By definition, mult n (z) = 0 if z is not an eigenvalue Notation dim n is used for the dimension of l (V n ) which is the same as the number of points in V n Proposition (i) If z / E(M 0,M), then mult n (z) = mult n (R(z)), () and every corresponding eigenfunction at depth n is an extension of an eigenfunction at depth n (ii) If z / σ(d), φ(z) = 0 and R(z) has a removable singularity at z, then mult n (z) = dim n, () and every corresponding eigenfunction at depth n is localized (iii) If z σ(d), both φ(z) and φ (z) have poles at z, R(z) has a removable singularity at z, and d dzr(z) 0, then mult n (z) = m n mult D (z) dim n +mult n (R(z)), () and every corresponding eigenfunction at depth n vanishes on V n (iv) If z σ(d), but φ(z) and φ (z) do not have poles at z, and φ(z) 0, then mult n (z) = m n mult D (z) + mult n (R(z)) (6) In this case m n mult D (z) linearly independent eigenfunctions are localized, and mult n (R(z)) more linearly independent eigenfunctions are extensions of corresponding eigenfunction at depth n (v) If z σ(d), but φ(z) and φ (z) do not have poles at z, and φ(z) = 0, then mult n (z) = m n mult D (z) + mult n (R(z)) + dim n (7) provided R(z) has a removable singularity at z In this case there are m n mult D (z)+dim n localized and mult n (R(z)) non-localized corresponding eigenfunctions at depth n

11 Vibration modes of n-gaskets and other fractals (vi) If z σ(d), both φ(z) and φ (z) have poles at z, R(z) has a removable singularity at z, and d dzr(z) = 0, then mult n (z) = mult n (R(z)), (8) provided there are no corresponding eigenfunctions at depth n that vanish on V n In general we have mult n (z) = m n mult D (z) dim n +mult n (R(z)) (9) (vii) If z / σ(d), φ(z) = 0 and R(z) has a pole z, then mult n (z) = 0 and z is not an eigenvalue (viii) If z σ(d), but φ(z) and φ (z) do not have poles at z, φ(z) = 0, and R(z) has a pole z, then mult n (z) = m n mult D (z) (0) and every corresponding eigenfunction at depth n vanishes on V n In the next theorem we establish the relation between eigenprojectors of spectrally similar operators Namely, we show how one can find the eigenprojector P n,z of M n corresponding to an eigenvalue z, if the eigenprojector P n,r(z) of M n corresponding to eigenvalue R(z) is known We state this theorem for n = and M = M, and the analogous relation holds for any n As before, we define φ (z) = φ(z)r(z) Theorem (i) In the case of Proposition (i), P z = φ(z) d dz R(z)(P 0 (D z) C)P 0,R(z) (P 0 B(D z) ) () (ii) In the case of Proposition (ii), P z = (P 0 (D z) C)(ψ 0 (z)m 0 ψ (z)) (P 0 B(D z) ) () where ψ 0 (x) = φ(x)/(z x) and ψ (x) = φ (x)/(z x) This implies, in particular, that there is an one-to-one map v 0 v = v 0 (D z) Cv 0 from l (V 0 ) onto the eigenspace of M corresponding to z (iii) In the case of Proposition (iii), the poles of φ(z) and φ are simple and so R(z) has a removable singularity at z, P z P D,z = P z and P 0 P z = 0, which means that the corresponding eigenfunctions of M vanish on V 0 Moreover, rankp D,z rankp z = rank(ψ 0 (z)m 0 ψ (z)i 0 ) = corankp 0,R(z) where ψ 0 (x) = φ(x)(z x) and ψ (x) = φ (x)(z x) In addition, the following relations hold P z = P D,z + ψ 0 (z) P D,zC(M 0 R(z)) (I 0 P 0,R(z) )BP D,z () and P D,z CP 0,R(z) = 0 Note that I 0 P 0,R(z) is the projector from l (V 0 ) onto the space, where (D z) is a well defined bounded operator

12 Vibration modes of n-gaskets and other fractals (iv) In the case of Proposition (iv), P z = P D,z + φ(z) d dz R(z)(P 0 (D z) C)P 0,R(z) (P 0 B(D z) ) () and the projector P D,z is orthogonal to the second term in the right hand side of this formula In particular, P z P D,z = P D,z (v) In the case of Proposition (v), P z is the sum of the right hand sides in () and () (vi) In the case of Proposition (vi), provided there are no corresponding eigenfunction at depth n that vanish on V n, we have P z = ψ(z) d dz R(z) (P 0 (D z) C)P 0,R(z) (P 0 B(D z) ) () In general, this formula is combined with (vii) In the case of Proposition (vii) we formally have P z = 0 (viii) In the case of Proposition (viii) we have P z = P D,z Proof Item (i) is the same as Theorem ; it is inserted here also for the sake of completeness To prove item (ii), we pass to the limit as x z in the formula, which can be re-written as (M x) = (D x) + + z x (P 0 (D x) C)(ψ 0 (x)m 0 ψ (x)) (P 0 B(D x) ) (6) Then the statements to be proved follow if we pass to the limit as x z in this formula To prove item (iii), we again pass to the limit as x z in formula () We see that P 0 P z 0 if and only if lim (x x z z) (ψ 0 (x)m 0 ψ (x)i 0 ) 0, d that is only possible if dz R(z) = 0 Therefore P 0P z = 0 in our case Relation () follows from () Note that ψ 0 (z)m 0 ψ (z)i 0 = P 0 MP D,z MP 0 if z σ(d) Hence rank(ψ 0 (z)m 0 ψ (z)i 0 ) = rank(p D,z P z ) In addition, we have that ψ 0 (z)m 0 ψ (z)i 0 is nonpositive Also we see that P 0 (M z) P 0 is a bounded operator on l (V 0 ) and so we have P 0 (M z) P 0 = lim(z x)(ψ 0 (x)m 0 ψ (x)i 0 ) Hence P 0 (M z) P 0 = 0 if x z and only if R(z) has a pole at z or R(z) ρ(m 0 ) If R(z) has a removable singularity at z then ψ 0 (z) d dz R(z)P 0(M z) P 0 = P 0 R(z) To prove item (iv), note that the relation P z P D,z = P D,z easily follows from the fact that φ and φ do not have poles Then, if we restrict everything to the orthogonal complement of the image of P D,z, we can apply item (i) of this theorem Item (v) follows from items (ii) and (iv) The proof of item (vi) is a combination of the proofs of items (i) and (iii) Items (vii) and (viii) easily follow from ()

13 Vibration modes of n-gaskets and other fractals Sierpiński gasket Spectral analysis on the Sierpiński gasket originates from physics papers [0, ] and is well known [7,, 6, 7] In this section we show how one can study it using our methods Note that recently Sierpiński lattices appeared as the Schreier graphs of so called Hanoi towers groups [9, 8, ] x x Figure The Sierpiński gasket and its V network Figure shows the depth one approximation to the Sierpiński gasket The depth Laplacian matrix M = M, which is obtained from the above figure, is M = The eigenfunction extension map is (D z) C = From these we have that and +(7 z)z ( +z) +z( 7+z) ( +z) +z( 7+z) φ(z) = x 6 x ( +z) +z( 7+z) +(7 z)z ( +z) +z( 7+z) z z + z R(z) = ( z)z The eigenvalues of M written with multiplicities are { σ(m) =,,,, },0 x x ( +z) +z( 7+z) ( +z) +z( 7+z) +(7 z)z and the corresponding eigenvectors are {-, -, 0, 0, 0, }, {-, 0, -, 0,, 0}, {0, -, -,, 0, 0}, {, 0, -, -, 0, }, {, -, 0, -,, 0}, {,,,,, } The eigenvalues of D written with multiplicities are { σ(d) =,, } and the corresponding eigenvectors are {-, 0, }, {-,, 0}, {,, } The equation ϕ = 0 has as its solution { } so the exceptional set is { E(M 0,M) =,, }

14 Vibration modes of n-gaskets and other fractals Figure The graph of R(z) for the Sierpiński gasket We can find the multiplicities of these exceptional values by using Proposition For the value, which is a pole of φ(z) and in σ(d), we use Proposition (iii) to find the multiplicities: mult ( ) = + = 0, mult ( ) = =, mult ( ) = 8 + =, For the value, which is also a pole of φ(z) and in σ(d), we again use Proposition (iii) to find the multiplicities: mult ( ) = + = 0, mult ( ) = 6 + = 0, mult ( ) = = 0, For the value, since / σ(d) and φ(z) = 0, we use Proposition (ii) to find the multiplicities Here the multiplicity of in the nth depth is equal to the dimension at depth n mult ( ) =, mult ( ) = 6, mult ( ) =, Table shows the ancestor-offspring structure of the eigenvalues of the Sierpiński gasket The symbol * indicates branches and ξ (z) = 6z 8 ξ (z) = + 6z 8

15 Vibration modes of n-gaskets and other fractals of the inverse function R (z) computed at the ancestor value z By Proposition (i) the ancestor and the offspring have the same multiplicity The empty columns represent exceptional values If they are eigenvalues of the appropriate M n, then the multiplicity is shown in the right hand part of the same row z σ(m 0 ) 0 mult 0 (z) z σ(m ) 0 mult (z) z σ(m ) 0 ξ ( ) ξ ( ) mult (z) 6 z σ(m ) 0 mult (z) 6 Table Ancestor-offspring structure of the eigenvalues on the Sierpiński gasket By induction one can obtain the following proposition, which is known in the case of the Sierpiński gasket (see [, 6, 7]) Notation R n A is used for the preimage of a set A under the n-th composition power of the function R Proposition (i) σ(m 0 ) = {0, } (ii) For any n 0 and for any n we have In particular, for n σ(m n ) n m=0 n σ(m n ) = { } ( n σ(m n ) = {0, } ( (iii) For any n 0, dim n = n+ + (iv) For any n 0, mult n (0) = m=0 (v) For any n 0, mult n ( ) = n + R m {0, } m=0 R m {0, } ) ) ( n R m { } m=0 R m { } )

16 Vibration modes of n-gaskets and other fractals 6 (vi) If z R k { } then mult n(z) = n k + for n, 0 k n (vii) If z R k { } then mult n(z) = n k for n, 0 k n Corollary The normalized limiting distribution of eigenvalues (the integrated density of states) is a pure point measure κ with the the set of atoms ) ( { } ( ) R m { } R m { } m=0 m=0 Moreover, κ ( { }) =, and if z R m {, } κ({z}) = m For the Sierpiński gasket we also demonstrate how one can compute the eigenprojectors for the two most interesting eigenvalues, z = and z = For the former case we use Theorem (ii) We compute ψ 0 ( ) = and ψ ( ) = and so P n+, (P = n ( ) )( ) D n Cn ( ( ) ) M n + P n B n Dn (7) For the case z = we use Theorem (iii) with R( ) = 0 and ψ 0( ) = and so P n+, = P D n, + P D n, C nm n B n P Dn, (8) Note that one can show that the term I n P n,0 is the projector to the orthogonal complement to constants and so can be omitted in this case Note also that P Dn, has a simple block structure with blocks and that D n has a block structure with blocks The matrices of C n and B n also have similarly simple block structure with block equivalent, depending on the labeling of vertices, to except for boundary vertices The computation of the eigenprojectors using Theorem (i) plays an important role in [7]

17 Vibration modes of n-gaskets and other fractals 7 6 A non-pcf analog of the Sierpiński gasket Several non-pcf analogs of the Sierpiński gasket were introduced in [0] Here we analyze the simplest one of them This fractal can be constructed as a self-affine fractal in R using 6 affine contractions, as shown in [0] It is finitely ramified but not pcf in the sense of Kigami Figure 6 shows the V network for this fractal x x x x 6 x 7 x x Figure 6 The non-pcf analog of the Sierpiński gasket and its V network The matrix of the depth- Laplacian M = M is M = and the eigenfunction extension map is (D z) C = +6z +6z 6 8z+z ( z+z ) ( z+z ) +6z +6z ( z+z ) 6 8z+z ( z+z ) +6z +6z ( z+z ) ( z+z ) 6 8z+z +6z +6z +6z Moreover, we compute that and φ(z) = z 7z + 8z z(z )(z ) R(z) = z The eigenvalues of D, written with multiplicites, are { σ(d) =,,, } with corresponding eigenvectors {-, -, -, }, {-, 0,, 0}, {-,, 0, 0}, {,,, } One can also compute { σ(m) =,,,,, },0

18 Vibration modes of n-gaskets and other fractals Figure 7 The graph of R(z) for the non-pcf analog of the Sierpiński gasket with the corresponding eigenvectors {-, -, -, 0, 0, 0, }, {-, -, -,,,, 0}, {-, 0,, -, 0,, 0}, {-,, 0, -,, 0, 0}, {, 0, -, -, 0,, 0}, {, -, 0, -,, 0, 0}, {,,,,,, } It is easy to see that φ(z) = 0 has one solution { E(M 0,M) = {,,, } Thus, the exceptional set is } z σ(m 0 ) 0 mult 0 (z) z σ(m ) 0 mult (z) z σ(m ) 0 mult (z) 6 7 Table Ancestor-offspring structure of the eigenvalues on the non-pcf analog of the Sierpiński gasket To begin the analysis of the exceptional values, note that is a pole of R(z) and therefore is not an eigenvalue by Proposition (vii) We are interested in the values of R(z) in the other exceptional points, which are R() = R( ) = 0 and R( ) = It is easy to see that and are poles of φ(z) and so we can use Proposition (iii)

19 Vibration modes of n-gaskets and other fractals 9 to compute the multiplicities We obtain and mult () = + = 0, mult ( ) = + = 0, mult () = 7 + = 6, mult ( ) = = Since is not a pole of φ(z), we can use Proposition (iv) to compute the multiplicities mult ( ) = + = and mult ( ) = 6 + = 7 The ancestor-offspring structure of the eigenvalues on the non-pcf analog of the Sierpiński gasketis shown in Table The symbol * indicates branches of the inverse function R (z) computed at the ancestor value The multiplicity of the ancestor is the same as that of the offspring by Proposition (i) The empty columns correspond to the exceptional values If they are eigenvalues of the appropriate M n, then the multiplicity is shown in the right hand part of the same row Theorem 6 (i) For any n 0 we have that σ( n ) n m=0 R m({0, }) and σ( ) = {0,,, } (ii) For n we have that n σ( n ) = {0, } ( m=0 ) ( n R m {, } + 6n (iii) For any n 0 we have dim n = (iv) For any n 0 we have mult n (0) = (v) For any n we have mult n ( ) = 6n + m=0 R m {,} ) (vi) For any n and z R n {, } we have that mult n(z) = (vii) For any 0 m n and z R m {, } we have that mult n (z) = mult n m ( ) = 6n m + (viii) For any 0 m n and z R m { } we have mult n( ) = 6n m 6 (ix) For any 0 m n and z R m {} we have mult n () = 6n m 6 Proof For this fractal we have σ( 0 ) = {0, } with mult 0( ) = and, for the purposes of Proposition, m = 6 Item (i) is obtained above in this section Item (ii) follows from the subsequent items Item (iii) is straightforward by induction Item (iv) follows from Proposition (i) because 0 is a fixed point of R(z)

20 Vibration modes of n-gaskets and other fractals 0 Item (v) easily follows from Proposition (iv) Items (vi) and (vii) follows from the items above Items (viii) and (ix) follows from Proposition (iii) because mult n ( ) = 6n mult n () = 6 n + 6n + 6n + 6 n + = 6n 6, + = 6n 6 Corollary 6 The normalized limiting distribution of eigenvalues (the integrated density of states) is a pure point measure κ with the the set of atoms where κ ( { }) = and m=0 R m {,}, κ({z}) = 6 m if z R m {, }; κ({z}) = 6 m if z R m { }; κ({z}) = 6 m if z R m {} 7 Level- Sierpiński gasket The level- Sierpiński gasket is shown in Figure 8 It had been used as an example in several works [, 0, 6, and references therein] In particular, the spectrum is computed in the recent paper [9] independently of our work The matrix for the depth- Laplacian M = M is M =

21 Vibration modes of n-gaskets and other fractals and the eigenfunction extension map (D z) C is +09z z +8z ( 6z+z )( z+6z ) +09z z +8z ( 6z+z )( z+6z ) +z ( +z z +6z ) 9+7z ( 6z+z )( z+6z ) 9+7z ( 6z+z )( z+6z ) +z ( +z z +6z ) 9+7z ( 6z+z )( z+6z ) +z ( +z z +6z ) +09z z +8z ( 6z+z )( z+6z ) +09z z +8z ( 6z+z )( z+6z ) +z ( +z z +6z ) 9+7z ( 6z+z )( z+6z ) +z ( +z z +6z ) 9+7z ( 6z+z )( z+6z ) 9+7z ( 6z+z )( z+6z ) +z ( +z z +6z ) +09z z +8z ( 6z+z )( z+6z ) +09z z +8z ( 6z+z )( z+6z ) 8z+z 8z+z 8z+z Moreover, we compute that and φ(z) = R(z) = (z )(6z 7) (z )(z )( 6z + z ) 6z(z )(z )(z ) 6z 7 x x x 7 x x 0 Figure 8 The level- Sierpiński gasketand its V network x x 9 x 6 x 8 x Figure 9 The graph of R(z) for the level- Sierpiński gasket The eigenvalues of D, written with multiplicities are { σ(d) =, ( + ),,,,, ( } )

22 Vibration modes of n-gaskets and other fractals One can also compute { σ(m) =,,,, ( + ), ( + ),, ( ), ( } ),0 We find that φ(z) = 0 has two solutions { 7 6 }, { } Thus, the exceptional set is { E(M 0,M) =, ( + ),,, ( ), 7 } 6 z σ(m 0 ) 0 mult 0 (z) z σ(m ) 0 ± ± mult (z) z σ(m ) 0 ± ± mult (z) 6 Table Ancestor-offspring structure of the eigenvalues on the level- Sierpiński gasket To begin the analysis of the exceptional values, note that find the poles of R(z) and see if it is an exceptional value It is easy to see that,, ( ) and (+ ) are poles of φ(z) and so we can use Proposition (iii) to compute the multiplicities We obtain mult ( ) = + = 0, mult ( ) = 0 + =, mult ( ) = + = 0, mult ( ) = 0 + =, mult ( ± ) = + = 0, mult ( ± ) = = 0 Note that R( ) = R( ) = 0 and R(± ) = Also, is not a pole of φ(z) but φ( ) = 0 and therefore we use Proposition (v) to compute the multiplicities We obtain mult ( ) = =, mult ( ) = = 6

23 Vibration modes of n-gaskets and other fractals The ancestor-offspring structure of the eigenvalues on the level- Sierpiński gasket is shown in Table The multiplicity of the ancestor is the same as that of the offspring by Proposition (i) The empty columns correspond to the exceptional values If they are eigenvalues of the appropriate M n, then the multiplicity is shown in the right hand part of the same row Theorem 7 (i) For any n 0 we have that σ( n ) n m=0 R m({0, }) and σ( ) = {, ( ± ),, } (ii) For n 0 we have that σ( n ) = (R n (0)) ( R (n ) ( (iii) For n 0 we have dim n = + 7 (6n ) ± (iv) For n 0 we have that mult n (0) = mult n () = )) { } (v) For n and for z R k (), 0 k we have that mult n (z) = (vi) For n 0 we have that mult n ( ) = 6n + 8 (vii) For n and 0 k n we have for z R k {, } that mult n (z) = (6n k ) Note as a special case k = 0 which gives the multiplicities of and (viii) For n with 0 k n we have that for z R k ( ± ) mult n (z) = mult n k ( ) = 6n k + 8 (ix) For any n with 0 k n we have that for z R k ( ± ) mult n (z) = 0 Proof For this fractal we have σ( 0 ) = {0, } with mult 0( ) = and, for the purposes of Proposition, m = 6 Item (i) is obtained above in this section Item (ii) follows from the subsequent items Item (iii) is straightforward by induction Item (iv) follows from Proposition (i) because 0 is a fixed point of R(z) and because R() = 0 Item (v) easily follows from Proposition (i) and Item (iv) Item (vi) follows from the previous items and Proposition (v) Item (vii) follows from Proposition (iii) Item (viii) follows from Proposition (i) Item (ix) follows from Proposition (iii), and as a consequence none of these values appear in the spectrum

24 Vibration modes of n-gaskets and other fractals Corollary 7 The normalized limiting distribution of eigenvalues (the integrated density of states) is a pure point measure κ with the the set of atoms { } ( m=0 R m {,, ± } ) Moreover, κ ( { }) = 7 and κ({z}) = 7 6 m if z R m {, }; κ({z}) = 7 6 m if z R m {± } 8 A fractal -tree The fractal tree is a fractal that is approximated by triangles as shown in Figure 0, but in the limit is a topological tree It appeared as the limit set of the Gupta-Sidki group, see [, 8, and references therein] x 6 x x 7 x x x x x 9 x 8 Figure 0 The fractal -tree and its V network The depth- Laplacian matrix M = M is M =

25 Vibration modes of n-gaskets and other fractals and the eigenfunction extension map (D z) C is +z( 7+z) 9 8z(6+z( 9+z)) ( +z) ( +z)(+z( +z)) ( +z) ( +z)(+z( +z)) 7+8( z)z ( +z)(+z( +z)) 9 8z(6+z( 9+z)) 9 8z(6+z( 9+z)) From here, we compute that ( +z) ( +z)(+z( +z)) +z( 7+z) 9 8z(6+z( 9+z)) ( +z) ( +z)(+z( +z)) 9 8z(6+z( 9+z)) 7+8( z)z ( +z)(+z( +z)) 9 8z(6+z( 9+z)) ( +z) ( +z)(+z( +z)) ( +z) ( +z)(+z( +z)) +z( 7+z) 9 8z(6+z( 9+z)) 9 8z(6+z( 9+z)) 9 8z(6+z( 9+z)) 7+8( z)z ( +z)(+z( +z)) and φ(z) = z 9 8z + 7z z R(z) = z(z )(z ) Figure The graph of R(z) for the fractal tree The eigenvalues of D written with multiplicities are { σ(d) =,, ( + ),,, ( ) } and the corresponding eigenvectors are {, 0,,, 0, }, {,, 0,,, 0}, {,,,,,}, {,0,,,0,}, {,,0,,,0}, and { +, +, +,,,} Computing the eigenvalues of M with multiplicities gives { σ(m) =,,,,,,, },0 and the corresponding eigenvectors are {0, 0,, 0, 0, 0, 0, 0, }, {0,, 0, 0, 0, 0, 0,, 0}, {, 0, 0, 0, 0, 0,, 0, 0}, {, 0,,, 0,, 0, 0, 0}, {,, 0,,, 0, 0,

26 Vibration modes of n-gaskets and other fractals 6 0, 0}, {,,,,,,,, }, {, 0,,, 0,,, 0, }, {,, 0,,, 0,,, 0}, {,,,,,,,, } The only solution of φ(z) = 0 is As such, the exceptional set is { E(M 0,M) =,, ( + ), ( } ) For analysis of exceptional values, one can find R(z) at each exceptional point by R (0) = {0, } {, and R ( ) =, ( ), ( + } ) Using Proposition, one can determine the multiplicities of the exceptional values For the value, which is a zero of φ(z), we use Proposition (v) to find the multiplicities mult ( ) = 0 () =, mult ( ) = () = 7 For the value, which is a pole of φ(z), we use Proposition (iii) to find the multiplicities mult ( ) = 0 () + = 0, mult ( ) = () 9 + = 0 For the values ( + ) and ( ), which are poles of φ(z), we use Proposition (iii) to find the multiplicities mult ( ( ± )) = 0 () + = 0, mult ( ( ± )) = () 9 + = 0 z σ(m 0 ) 0 mult 0 (z) z σ(m ) 0 ± mult (z) z σ(m ) 0 ± mult (z) 7 Table Ancestor-offspring structure of the eigenvalues of the fractal tree The ancestor-offspring structure of the eigenvalues of the Fractal Tree is shown in Table As before, the symbol * indicates branches of the inverse function R (z) computed at the ancestor value The multiplicity of the ancestor equals that of the offspring by Proposition (i) The exceptional values are represented by the empty columns If they are eigenvalues of the appropriate M n, then the multiplicity is shown in the right hand part of the same row

27 Vibration modes of n-gaskets and other fractals 7 Theorem 8 (i) For any n 0 we have that σ( n ) n m=0 R m({0, }) { } and σ( ) = {, ( ± ), } (ii) For n we have that σ( n ) = And for n = we have σ( ) = {0,,, } (iii) For n 0 we have dim n = + ( n ) { 0, } ( n { } ) R k, k=0 (iv) For n 0 we have mult n (0) = mult n () = (v) For n with 0 k n we have that if z R k () then mult n (z) = mult n k () = (vi) For n 0 we have that mult n ( ) = n + (vii) For n with 0 k n we have for z R k ( ) that mult n (z) = mult n k ( ) = mult n k ( ) = n k + (viii) For n we have mult n ( ) = 0 (ix) For n with 0 k n we have that if z R k ( ± ) then mult n (z) = 0 Proof For this fractal we have σ( 0 ) = {0, } with mult 0( ) = and, for the purposes of Proposition, m = 6 Item (i) is obtained above in this section Item (ii) follows from the subsequent items Item (iii) is straightforward by induction Item (iv) follows from Proposition (i) because 0 is a fixed point of R(z) and because R() = 0 Item (v) easily follows from Proposition (i) and Item (iv) Item (vi) follows from the previous items and Proposition (v) Item (vii) follows from Proposition (i) Items (viii) and (ix) follow from Proposition (iii), and as a consequence none of these values appear in the spectrum Corollary 8 The normalized limiting distribution of eigenvalues (the integrated density of states) is a pure point measure κ with the the set of atoms { } ( { R m } ) m=0 Moreover, κ ( { }) =, and κ({z}) = m if z R m { }

28 Vibration modes of n-gaskets and other fractals 8 Figure The hexagasket and its V network x x x 0 x 8 x x 7 x x x x 9 x x 6 9 Hexagasket The hexagasket, or the hexakun, is a fractal which in different situations [, 6,, 6, 0,,, and references therein] is called a polygasket, a 6-gasket, or a (,, )-gasket The depth- approximation to it is shown in Figure The matrix of the depth- Laplacian M = M is M = and the eigenfunction extension map (D z) C is +z(+z( 9+z)) ( 6z+z )(7+8z( +z)) +z ( 6z+z )(7+8z( +z)) +(7 z)z ( 6z+z )(7+8z( +z)) +z(+z( 9+z)) ( 6z+z )(7+8z( +z)) +(7 z)z ( 6z+z )(7+8z( +z)) +z ( 6z+z )(7+8z( +z)) +(7 z)z ( 6z+z )(7+8z( +z)) +z(+z( 9+z)) ( 6z+z )(7+8z( +z)) +z ( 6z+z )(7+8z( +z)) +z ( 6z+z )(7+8z( +z)) +z(+z( 9+z)) ( 6z+z )(7+8z( +z)) +(7 z)z ( 6z+z )(7+8z( +z)) +z ( 6z+z )(7+8z( +z)) +(7 z)z ( 6z+z )(7+8z( +z)) +z(+z( 9+z)) ( 6z+z )(7+8z( +z)) +(7 z)z ( 6z+z )(7+8z( +z)) +z ( 6z+z )(7+8z( +z)) +z(+z( 9+z)) ( 6z+z )(7+8z( +z)) +( z)z +( z)z ( 6z+z )(7+8z( +z)) ( 6z+z )(7+8z( +z)) ( 6z+z )(7+8z( +z)) +( z)z +( z)z ( 6z+z )(7+8z( +z)) ( 6z+z )(7+8z( +z)) ( 6z+z )(7+8z( +z)) +( z)z +( z)z ( 6z+z )(7+8z( +z)) ( 6z+z )(7+8z( +z)) ( 6z+z )(7+8z( +z)) Moreover, we compute that φ(z) = + (z )z (z + 6z ) (7 + 8z(z ))

29 Vibration modes of n-gaskets and other fractals 9 and R(z) = z(z )(7 z + 6z ) z Figure The graph of R(z) for the hexagasket The eigenvalues of D, written with multiplicities, are { σ(d) =,,, ( ± ), ( ± ), ( ± ) } One can also compute σ(m) = {,,,,,,,,,, },0 with the corresponding eigenvectors {0,, 0, 0, 0, 0,, 0, 0, 0, 0, }, {, 0, 0, 0,, 0, 0, 0, 0, 0,, 0}, {, 0, 0,, 0, 0, 0, 0, 0,, 0, 0}, {, 0,,, 0, 0, 0, 0,, 0, 0, 0}, {0,,, 0, 0, 0,,, 0, 0, 0, 0}, {,, 0, 0,,, 0, 0, 0, 0, 0, 0}, {,,, 0, 0, 0, 0, 0, 0,,, }, {,, 0, 0,,, 0,,,, 0, }, {0,,,,, 0,,, 0,,, 0}, {,, 0,,,,,,,, 0, }, {0,,,,,,,,,,, 0}, {,,,,,,,,,,, } Thus, the exceptional It is easy to see that φ(z) = 0 has two solution and set is { E(M 0,M) =, ( ± ), ( ± ), To begin the analysis of the exceptional values, note that is the pole of R(z) and therefore is not an eigenvalue by Proposition (vii) It is easy to see that ( ± ( ) ) and ± are the four poles of φ(z) and so we can use Proposition (iii) to compute the multiplicities We obtain mult ( ( ± )) = = 0, mult ( ( ± )) = 6 + =, mult ( ( ± )) = = 0, mult ( ( ± )) = = 0 }

30 Vibration modes of n-gaskets and other fractals 0 z σ(m 0 ) 0 mult 0 (z) z σ(m ) 0 ± ± mult (z) 6 z σ(m ) 0 ± ± ± mult (z) Table Ancestor-offspring structure of the eigenvalues on the hexagasket The exceptional value is in the spectrum σ(d), not a pole of φ(z) and φ( ) = 0 For this reason we can use Proposition (v) to compute the multiplicities mult ( ) = = 6, mult ( ) = = 0 As in the other sections, the multiplicities of all eigenvalues at depths 0, and are shown in Table Theorem 9 (i) σ(m 0 ) = {0, { } (ii) We have that σ(m ) = 0,,,, } and for n we have { σ(m n ) = 0, } ( n { R m,, } ) ( { n ± }) R m m=0 m=0 (iii) For any n 0 we have dim n = n (iv) For any n 0, mult n (0) = and mult n ( ) = 6 + 6n (v) For any n and 0 k < n we have that if x R k () then mult n (z) = (vi) For any n and 0 k < n we have that if z R k {, } then mult n (z) = 6 + 6n k (vii) For any n and 0 k < n we have that if z R k ( ± ) then mult n (z) = 6n k (viii) For n 0 we have mult n ( ± ) = 0

31 Vibration modes of n-gaskets and other fractals Proof For this fractal we have σ( 0 ) = {0, } with mult 0( ) = and, for the purposes of Proposition, m = 6 Item (i) is obtained above in this section Item (ii) follows from the subsequent items Item (iii) is straightforward by induction Item (iv) follows from Proposition (i) because 0 is a fixed point of R(z), and from Proposition (v) Items (v) and (vi) follow from Proposition (i) Items (vii) and (viii) follow from Proposition (iii) Corollary 9 The normalized limiting distribution of eigenvalues (the integrated density of states) is a pure point measure κ with the the set of atoms { } ( { R m,, ± }) Moreover, κ ( { }) = 9, and m=0 κ({z}) = 9 6 m if z R m {, }; κ({z}) = 9 6 m if z R m { ± } 0 One dimensional interval as a self-similar set In this section we show how our results allow us to recover classically known information about the spectrum of the discrete Laplacians that approximate the usual one dimensional continuous Laplacian The unit interval [0,] can be represented as a self-similar set in various ways Here we consider three cases: when it subdivided into two, three or four subintervals of equal length In our notation this means that m is,, or The depth- networks for these cases are shown in Figure The first two cases were also discussed in [9] Note that in each case the function R(z) is the same as the Chebyshev polynomial of degree m for the interval [0,], which is the smallest interval that contains the spectrum of the matrices M n It is shown in [9], in particular, that the iterations of these polynomials are related in a natural way with the Riemann zeta function x x x x x x x x x x x x Figure V networks for the interval in cases m =,, respectively Case m = The matrix of the depth- Laplacian M = M is 0 M = 0

32 Vibration modes of n-gaskets and other fractals and the eigenfunction extension map is ( (D z) C = (z ) (z ) ) Moreover, we compute that and φ(z) = ( z) R(z) = z( z) The only eigenvalue of D is σ(d) = {} One can also compute σ(m) = {,,0} with the corresponding eigenvectors{{-, -, }, {-,, 0}, {,, }} It is easy to see that φ(z) 0 Thus, the exceptional set is E(M 0,M) = {} To begin the analysis of the exceptional value, note that R(z) does not have any poles We are interested in the value of R(z) at the exceptional point, which is R() = It is easy to see that is a pole of φ(z), R(z) has a removable singularity at z, and d dzr(z) = 0 So for all n we can use Proposition (vi) to compute its multiplicity mult n () = Figure The graph of R(z) for F = [0, ] with m =, m = and m = respectively Case m = The matrix of the depth- Laplacian M = M is 0 0 M = and the eigenfunction extension map is Moreover, we compute that (D z) C = ( (z ) 8z+z 8z z (z ) 8z z 8z+z φ(z) = ( ) z )( z )

33 Vibration modes of n-gaskets and other fractals and R(z) = z( z) The eigenvalues of D, written with multiplicities, are { σ(d) =, } with corresponding eigenvectors{{-, }, {, }} One can also compute σ(m) = {,, },0 with the corresponding eigenvectors{{, -, -, }, {-, -,, }, {-,, -, }, {,,, }} It is easy to see that φ(z) 0 Thus, the exceptional set is { E(M 0,M) =, } Again, note that R(z) does not have any poles We are interested in the values of R(z) in the exceptional points, which are R( ) = 0, R( ) = Since d dzr(z) = 0 in these points, we can use Proposition (vi) to obtain for all n mult n ( ) = mult n( ) = Case m = The matrix of the depth- Laplacian M = M is M = and the eigenfunction extension map is We compute that and (D z) C = 8z+z ( +z 6z +z ) 8z+z φ(z) = ( +z 6z +z ) 0z + z 8z R(z) = 8z(z )( z) ( +z 6z +z ) 8z+z 8z+z ( +z 6z +z ) The eigenvalues of D, written with multiplicities, are { ( σ(d) = + ),, ( ) }

34 Vibration modes of n-gaskets and other fractals with corresponding eigenvectors {{, } {,, {,0,},, }}, One can also compute σ(m) = {, ( + ),, ( ) },0 It is easy to see that φ(z) 0 Thus, the exceptional set is { ( E(M 0,M) = + ) (,, )} To begin the analysis of the exceptional values, note that R(z) does not have any poles We are interested in the values of R(z) at the exceptional points, which are R( ( + )) =, R() = 0, R( ( )) = d Once again, dzr(z) = 0 at these points, and by Proposition (vi) we have for all n Diamond fractal mult n ( ( + )) = mult n () = mult n ( ( )) = The diamond fractal is shown in figure 6 The diamond self-similar hierarchical lattice appeared as an example in several physics works, such as [] Recently the critical percolation on the diamond fractal was analyzed in [] x x x x Figure 6 The diamond fractal and its V network We can use the results obtained for the unit interval [0,] in Section 0, case m =, to develop the spectral decimation method for the diamond fractal The matrix of the depth- Laplacian M = M is M =

35 Vibration modes of n-gaskets and other fractals and the eigenfunction extension map is now the square matrix with the same entries ( ) (D z) C = (z ) while the functions and φ(z) = ( z) R(z) = z( z) are the same as for the unit interval, σ(d) = {,} has multiplicity two, and σ(m) = {,,,0} with the corresponding eigenvectors {-, -,, }, {-,, 0,0}, { 0,0,-, }, {,,,} The exceptional set is E(M 0,M) = {} Theorem (i) For any n 0 we have that σ( n ) = n m=0 R m ({0,}) (ii) For any n 0 we have dim n = + ( n ) (iii) For any n 0 we have mult n (0) = mult n () = (iv) For any n and 0 k n we have mult n (z) = n k + if z R k () Proof Item (i) follows from (iii) and (iv) Item (ii) is obtained by induction Item (iii) follows from Proposition (i), and the fact that R(0) = R() = 0 For the analysis of the only exceptional value z =, note that it is a pole of φ(z), R() =, R(z) has a removable singularity at, and d dzr() = 0 Therefore by Proposition (vi) we have for all n This implies Item (iv) mult n () = n n + + = n + Corollary The normalized limiting distribution of eigenvalues (the integrated density of states) is a pure point measure κ with the the set of atoms and κ ({z}) = m if z R m {} Acknowledgments m=0 R m {} The last author is very grateful to Rostislav Grigorchuk, Volodymyr Nekrashevych, Peter Kuchment and Robert Strichartz for many useful remarks and suggestions This research is supported in part by the NSF grant DMS-006

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