The spectral decimation of the Laplacian on the Sierpinski gasket

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1 The spectral decimation of the Laplacian on the Sierpinski gasket Nishu Lal University of California, Riverside Fullerton College February 22, 2011

2 1 Construction of the Laplacian on the Sierpinski gasket Find a sequence of graphs Γ m Define the graph Laplacian m on each Γ m The Laplacian on SG is = lim m c m m 2 The spectrum of the Laplacian on SG Outline for decimation method The eigenvalue diagram 3 The spectral zeta function The spectral decimation of the Laplacian on the Sierpinski February gasket 22, / 27

3 What is the Sierpinski gasket, SG? The spectral decimation of the Laplacian on the Sierpinski February gasket 22, / 27

4 Interesting facts about SG The dimension of the gasket is log 3 log The area of SG is 0. The boundary is of infinite length (the circumference is infinity). It is connected. The spectral decimation of the Laplacian on the Sierpinski February gasket 22, / 27

5 The construction of the Sierpinski gasket We study calculus on fractals by the approach of using discrete approximations. Consider three contraction mappings for SG: F i (x) = 1 2 (x q i) + q i for i = 0, 1, 2 where {q 0, q 1, q 2 } are the vertices of a triangle. The spectral decimation of the Laplacian on the Sierpinski February gasket 22, / 27

6 Let S(R 2 ) be the space of nonempty compact subsets of R 2. S(R 2 ) is a complete metric space with Hausdorff metric. Hausdorff metric The Hausdorff metric is a distance function on S(R 2 ): d H (A, B) = inf{ɛ > 0 : B A ɛ, A B ɛ } where A ɛ = {x R 2 : d(x, A) ɛ}. The spectral decimation of the Laplacian on the Sierpinski February gasket 22, / 27

7 Define F : S(R 2 ) S(R 2 ) by F (A) = F 0 (A) F 1 (A) F 2 (A). F is a contraction mapping on S(R 2 ). Definition The Sierpinski gasket, SG S(R 2 ) is a unique fixed point of F or SG = that is, SG is a self-similar set. 2 F i (SG) = F (SG) i=0 The spectral decimation of the Laplacian on the Sierpinski February gasket 22, / 27

8 The graph approximation Γ m SG The Sierpinski gasket is approximated by a sequence of graphs. Definition Let Γ 0 be the graph on V 0. Then we define Γ m := 2 F i (Γ m 1 ). i=0 The spectral decimation of the Laplacian on the Sierpinski February gasket 22, / 27

9 The spectral decimation of the Laplacian on the Sierpinski February gasket 22, / 27

10 Let Γ 0 be the graph on V 0 where V 0 = {q 0, q 1, q 2 }. Then the vertices of Γ m are obtained inductively by Define V = m>0 V m. V is dense in SG. V m := 2 F i (V m 1 ). i=0 So it is enough to use the sequence of finite sets V m to approximate SG. The spectral decimation of the Laplacian on the Sierpinski February gasket 22, / 27

11 The graph Laplacian on Γ m Definition Points x and y are called neighbors in Γ m if there is an m-cell containing both or x m y. Definition Let u : V m R be a function. We define the graph Laplacian on Γ m by where x V m V 0. m u(x) := x m y (u(x) u(y)) The spectral decimation of the Laplacian on the Sierpinski February gasket 22, / 27

12 Goal: define on SG from m on Γ m We define the pointwise formulation for the Laplacian on SG as the renormalized limit of m u(x) := lim m c m m u(x) Now the goal is to find the renormalization factor c m. The spectral decimation of the Laplacian on the Sierpinski February gasket 22, / 27

13 We use the concept of graph energy to define the Laplacian Graph Energy on Γ m SG Let u : V m R be a function. The graph energy of u is where r = 3 5. E m (u) = r m x m y (u(x) u(y)) 2 For any continuous function u on SG, E m (u Vm ) is a monotone increasing sequence of m. Graph energy on SG E(u) = lim r m (u(x) u(y)) 2 m x y m The spectral decimation of the Laplacian on the Sierpinski February gasket 22, / 27

14 To explore the relationship between E m (u) and E m+1 (u), we consider the following: Given u := V m R, extend u to V m+1. There are many extensions, but there is one unique extension ũ that minimizes E m+1 (u). ũ is called the harmonic extension. The harmonic extension satisfies rule at the new points V m+1 V m. Then we get E m+1 (ũ) = E m (u) for u. The spectral decimation of the Laplacian on the Sierpinski February gasket 22, / 27

15 Formulation of the Laplacian on SG To define the Laplacian from the graph energy requires the choice of a self-similar measure µ. Weak formulation of the Laplacian µ Let u dome and f C(SG). Then u dom µ and µ u = f if E(u, v) = fvdµ v dome vanishing on the boundary points. SG The spectral decimation of the Laplacian on the Sierpinski February gasket 22, / 27

16 finally, the Laplaican... We can derive the pointwise formulation of the Laplacian from the weak formulation. The Laplacian on SG is µ u(x) = 3 2 lim m 5m m u(x). The spectral decimation of the Laplacian on the Sierpinski February gasket 22, / 27

17 The normalized discrete Laplacian From now on, we will discuss the normalized discrete Laplacian. It is defined as follows: for u : V m R with u vanishing on V 0, the normalized Laplacian on Γ m is where x V m V 0. m u(x) := 1 (u(x) u(y)) 4 x m y The spectral decimation of the Laplacian on the Sierpinski February gasket 22, / 27

18 The spectrum of the Laplacian on SG Obtain solutions of the eigenvalue equation u = λu on SG as limits of solutions of the discrete version on V m V 0. m u m = λ m u m Construct λ m+1 from λ m via a polynomial R(z). The spectral decimation of the Laplacian on the Sierpinski February gasket 22, / 27

19 Outline of the decimation method If u is an eigenfunction of m+1 with eigenvalue λ, that is, m+1 u = λu and λ / B, then m (u Vm ) = R(λ)u Vm where B = { 5 4, 1 2, 3 2 } is the set of forbidden eigenvalues. If m u = R(λ)u and λ / B then there exists a unique extension ũ of u such that m+1 (ũ) = λ(ũ). The spectral decimation of the Laplacian on the Sierpinski February gasket 22, / 27

20 The special polynomial R(z) The decimation method relates the eigenvalues of successive graph Laplacians by the polynomial R(z) = z(5 4z). Given the eigenvalues of 0 to be {0, 3 2 }. To obtain eigenvalues of 1, we consider the inverse images of R(z): R (z) = z 8. R + (z) = z 8. The spectral decimation of the Laplacian on the Sierpinski February gasket 22, / 27

21 The eigenvalue diagram Operators Eigenvalues 0 0 [1] 3 2 [2] 1 0 [1] [2] [3] [1] 5 4 R + (3/4) R (3/4) [2] [2] 3 4 [3] [6] 5 [1] [1] 5 4 R + R(3/4) + R + [2] [2] R(3/4) + R(3/4) [3] [3] [6] 2 R(5/4) + R(5/4) [1] [1] 3 2 [15] 5 4 [4] 4 0 [1] 5 4 R(3/4) R(3/4) + [6] [6] 3 4 [15] 1 2 R(5/4) R(5/4) + [4] [4] The spectral decimation of the Laplacian on the Sierpinski February gasket 22, / 27

22 Eigenvalues of the Laplacian µ Every eigenvalue of µ has a form 5 i lim m 5m (R ) m j R ω (z 0 ) where ω is a word with ω = j taking values in {, +} and (R ) m j is the (m-j)th iteration power of R and z 0 = 3 4, 5 4. If R (z) < z then R n (z) is decreasing so it converges to a fixed point of R which is 0. The limit exists if R is applied after finite steps. The spectral decimation of the Laplacian on the Sierpinski February gasket 22, / 27

23 The spectral zeta function Definition Suppose A is a non-negative self-adjoint operator with discrete spectrum and positive eigenvalues {λ j }, then its spectral zeta function is ζ A (s) = λ j 0(λ j ) s/2. Example Consider the Dirichlet Laplacian on [0,1], A = d2 dx 2. Eigenvalues: λ = π 2 j 2, j = 1,2,... The spectral zeta function is ζ A (s) = j=1 (π2 j 2 ) s/2 = π s j=1 j s = π s/2 ζ(s). The spectral decimation of the Laplacian on the Sierpinski February gasket 22, / 27

24 The spectral zeta funtion associated to SG Alexander Teplyaev discovered the product structure of the spectral zeta function of the Laplacian on SG that involves a geometric part and a new zeta function associated with the polynomial R(z). This zeta function has many interesting properties... A similar product structure of the spectral zeta function was studied earlier by Michel Lapidus for fractal strings and can now be viewed as a special case of this situation. The spectral decimation of the Laplacian on the Sierpinski February gasket 22, / 27

25 Fractal String Example A fractal string L is a disjoint collection of open intervals of length l k. Define its geometric zeta function to be ζ L (s) = (λ k ) s. k=1 Let L = I k with I k = l k and let A = d2 be the Dirichlet Laplacian on dx 2 L. σ(a) = k=1 σ(a; I k) = { π2 n 2 : n 1, k 1}. l 2 k The spectral zeta function of A: ζ A = k,n 1 ( π2 n 2 ) s/2 = π s l 2 k=1 ls k n=1 n s. k Theorem Lapidus: ζ A (s) = π s ζ L (s)ζ(s), where ζ(s) is the Riemann zeta function. The spectral decimation of the Laplacian on the Sierpinski February gasket 22, / 27

26 Current research In order to generalize the decimation method for a certain class of fractals, Sabot introduced a function of several complex variables called the renormalization map. It is interesting to study the spectral zeta function of other self-similar sets in similar setting, in particular, its product structure in terms of the zeta function associated with the renormalization map. Thus far, jointly with Michel Lapidus, we have looked at the Sturm-Liouville operator of the form d d dµ dx on the half real line (blow-up of a unit interval). We established the spectal zeta function in terms of a renomalization map induced from the decimation method discovered by Sabot. The dynamics of an invariant curve of the renormalization map is used to compute the spectrum of the operator. We define the zeta function ζ f associated with the renormalization map f and relate it to the spectral zeta function ζ sp of the Sturm-Liouville operator via a product formula with a hyperfunction. The spectral decimation of the Laplacian on the Sierpinski February gasket 22, / 27

27 Fukushima M, Shima T., On the spectral analysis for the Sierpinski gasket Potential Analysis 1, (1992) Kigami J., Harmonic calulus on p.c.f self-similar sets. Trans. Am. Math. Soc. 335, (1993) Kigami J.,Lapidus M.L., Weyl s problem for the spectral distribution of Laplacian on p.c.f self-similar fractals Comm. Math. Phys. 158(1), (1993) Rammal R., Spectrum of harmonic exciations on fractals J. de Physique 45, (1984) Sabot C., Integrated density of states of self-similar Sturm-Liouville operators and holomorphic dynamics in higher dimension, Ann. I. H. Poincare, 37(3) (2001). R. Strichartz, Differential equations on fractals: a tutorial, Pincenton University Press, A. Teplyaev, Spectral zeta function of fractals and the complex dynamics of polynomials; Trans. Amer. Math. Soc., 359(2006), The spectral decimation of the Laplacian on the Sierpinski February gasket 22, / 27