Spectral Decimation for Families of Laplacians on the Sierpinski Gasket

Transcription

1 Spectral Decimation for Families of Laplacians on the Sierpinski Gasket Seraphina Lee November, 7 Seraphina Lee Spectral Decimation November, 7 / 53

2 Outline Definitions: Sierpinski gasket, self-similarity, Laplacian, etc. Spectral decimation method Generalization of spectral decimation Application: analogue of trigonometric functions on unit Interval Application: solutions to heat and wave equations on fractals Seraphina Lee Spectral Decimation November, 7 / 53

3 Introduction Unit Interval and Sierpinski Gasket Let s take a look at two standard fractals: unit Interval (I ) and Sierpinski Gasket (SG) x w w x x w Seraphina Lee Spectral Decimation November, 7 3 / 53

4 Definitions Iterated Function System (IFS): a system of contraction mappings used to define self-similar fractals: For the unit Interval, For SG, {F i : F i (x) = x + i, i =, } {F i : F i (x) = 3 (x q i) + q i, i =,, } where V = {q, q, q } denotes the 3 vertices of the triangle. Seraphina Lee Spectral Decimation November, 7 4 / 53

5 Definitions With these IFS s, we can come up with a self-similar definition for both fractals: I = F (I ) F (I ) SG = F (SG) F (SG) F (SG) Seraphina Lee Spectral Decimation November, 7 5 / 53

6 Definitions Define graph approximations inductively; V = {q i } V m = F am... F a (V ) Γ m is formed by joining images of F i Γ m together where F i q j = F i q j with q j q j, q j, q j V m. Γ m is the mth level graph approximation Seraphina Lee Spectral Decimation November, 7 6 / 53

7 (Some) Motivation What if we want to solve certain differential equations on a different spaces rather than R n? Example Heat Equation: Wave Equation: u(x, t) = x u(x, t) t u(x, ) = f (x) u(x, t) t = x u(x, t) u(x, ) = u u(x, ) = f (x) t Seraphina Lee Spectral Decimation November, 7 7 / 53

8 What is the Laplacian? You may remember from calculus that on R n, f = n f x i= i The Laplacian is an analogue of the second derivative, defined on the mth level approximation by the pointwise formula m f (x) = (f (x) f (y)) x my where x m y mean x and y are neighboring points on the mth level graph approximation, and on the fractals SG and I by f (x) = lim m r (m) ψ x (m) dµ ( mf (x)) Seraphina Lee Spectral Decimation November, 7 8 / 53

9 The Spectrum of the Laplacian Goal: We want to find all eigenvalue and eigenfunction pairs λ R, f : K R, such that for all x K and f (x) = λf (x) f (x) = for x K, the boundary; we say that such a function f satisfies Dirichlet boundary conditions. To do this, we can first look at the spectrum of the Laplacians on the mth level approximations of K, and take the limit and scale by the appropriate factor. Seraphina Lee Spectral Decimation November, 7 9 / 53

10 Spectral Decimation Outline We aim to describe the spectra of the laplacian without handling the limiting structure. Instead, we will look at the spectra for graph approximations. Given f (x) = λ m f (x) x V m Can we extend f and assign values for f on V m+ such that f (x) = λ m+ f (x) x V m+ Seraphina Lee Spectral Decimation November, 7 / 53

11 Spectral Decimation on the Interval We want new eigenfunctions on the (m + )th level to satisfy m+ f (z) = (f (z) f (y ) f (y )) where z y, z y and y, y V m. Seraphina Lee Spectral Decimation November, 7 / 53

12 Spectral Decimation on the Interval Eigenfunctions satisfy m+ f (z) = λ m+ f (z) = f (z) f (y ) f (y ) y z y Seraphina Lee Spectral Decimation November, 7 / 53

13 Spectral Decimation on the Interval We can solve for f (z) in terms of f (y ), f (y ), λ m+ : if λ m+. λ m+ f (z) = f (z) f (y ) f (y ) f (z) = (f (y ) + f (y )) λ m+ y z y Seraphina Lee Spectral Decimation November, 7 3 / 53

14 Eigenvalue Extension We now need to obtain λ m+ and confirm that the new function on V m+ is still an eigenfunction on V m. I.e. we know with neighbors y, y, and we want m f (y ) = λ m f (y ) with neighbors z, z m+ f (y ) = λ m+ f (y ) y z y z y (m)-cell (m)-cell Seraphina Lee Spectral Decimation November, 7 4 / 53

15 Spectral Decimation on the Interval Solving the system m f (y ) = λ m f (y ) = f (y ) f (y ) f (y ) m+ f (y ) = λ m+ f (y ) = f (y ) f (z) f (z ) yields the quadratic equation λ m+ = λ m (4 λ m ) and λ m+ = ± 4 λ m Seraphina Lee Spectral Decimation November, 7 5 / 53

16 Forbidden Eigenvalues Recall that we assumed λ m+. We handle this separately: For m =, by inspection, ( ) f = works. For m >, we find that restricting to V m+ works. f (x) = sin ( m πx) Seraphina Lee Spectral Decimation November, 7 6 / 53

17 Done? Eigenfunctions are linearly independent of each other, so there can be a maximum of #V m eigenfunctions at any given level m. On level m =, λ = works, as shown before. On any subsequent level m, take the m eigenfunctions, apply spectral decimation to obtain m eigenfunctions on level m, plus the function associated with λ = to obtain m eigenfunctions total. Seraphina Lee Spectral Decimation November, 7 7 / 53

18 Eigenfunctions m=5 p=.5 λ = m=5 p=.5 λ 3 = m=5 p=.5 λ 5 = m=5 p=.5 λ = m=5 p=.5 λ 4 = m=5 p=.5 λ 6 = Figure: Eigenfunctions on level m = 5 with lowest 6 eigenvalues Seraphina Lee Spectral Decimation November, 7 8 / 53

19 Where to from here? Does this process generalize to other self-similar spaces? Seraphina Lee Spectral Decimation November, 7 9 / 53

20 Solutions for Sierpinski Gasket Using the same process: Solve to get λ m+ f (w ) = 4f (w ) f (x ) f (w ) f (w ) f (x ) λ m+ f (w ) = 4f (w ) f (x ) f (w ) f (w ) f (x ) λ m+ f (w ) = 4f (w ) f (x ) f (w ) f (w ) f (x ) given λ m+, 5, 6 f (y ) = (4 λ m+)(f (x ) + f (x ) + f (x )) ( λ m+ )(5 λ m+ ) x w w x x w Seraphina Lee Spectral Decimation November, 7 / 53

21 Solutions for Sierpinski Gasket And solve λ m f (x ) = 4f (x ) f (x ) f (x ) f (y ) f (x ) λ m+ f (x ) = 4f (x ) f (w ) f (w ) f (w ) f (w ) to get given λ m+, 5, 6 λ m+ = 5 ± 5 4λ m y w x y w w w x x Seraphina Lee Spectral Decimation November, 7 / 53

22 Forbidden eigenvalues We handle the cases where λ m+ =, 5, 6 and find (by an inductive counting argument) that on every level m, { if m M m () = if m = M m (5) = 3m + 3 M m (6) = 3m 3 Seraphina Lee Spectral Decimation November, 7 / 53

23 Eigenfunctions m=3 r= λ =6.854 m=3 r= λ = m=3 r= λ 3 = m=3 r= λ 4 = Figure: Eigenfunctions on level m = 3 with lowest 4 eigenvalues Seraphina Lee Spectral Decimation November, 7 3 / 53

24 Constructing a one-parameter family of Laplacians The original pointwise formula for the Laplacian satisfies E(f, v) = ( f )vdµ for all v dom E an analogue of integration by parts, where Definition The bilinear form of energy on pairs of functions u, v on the mth level graph approximation is defined E m (u, v) = (u(x) u(y))(v(x) v(y)) x my x,y V m Seraphina Lee Spectral Decimation November, 7 4 / 53

25 Resistance We can always extend u on V m to V m+ such that E m+ (u, u) will be minimal among all extensions; this is called the harmonic extension. For harmonic functions extensions ũ of u, E m+ (ũ, ũ) = re m (u, u) for some constant r, so it makes sense for us to define energy on K by E(u, v) = lim m r m E m (u, v) We call r the renormalization factor. In the standard case, we used this as our resistance on the edges of the graph approximations of K. Seraphina Lee Spectral Decimation November, 7 5 / 53

26 Laplacian definition, revisited Definition If f is a function on K, f is the function that satisfies E(f, v) = ( f )vdµ for all v dom E The weak formulation above leads to a pointwise formula µ f (x) = lim m (x) ψ m dµ for the appropriate tent function ψ x my R (m) (f (x) f (y)) (x, y) Seraphina Lee Spectral Decimation November, 7 6 / 53

27 Generalizing... So far, all the calculations done were based on measure and resistance being uniform across K. What happens if we vary the measure and resistance in a symmetric, self-similar way? Seraphina Lee Spectral Decimation November, 7 7 / 53

28 Unit Interval, Parametrized Let measure and resistance distributed according to the parameters p and q p q 4 p q p q 3 4 p q p p q q 6 p p q q 8 p p q q 3 6 p p q q Then we want to use a modified IFS: {F i : F i (x) = x 4 + i, i =,,, 3} 4 Seraphina Lee Spectral Decimation November, 7 8 / 53

29 Special Properties of p + q = A common property of fractal laplacians is scaling { pq (u F i ) = 4 (u) F i if i =, 3 ( p)( q) 4 (u) F i if i =, If we want this renormalization factor to be constant across the interval, we need pq 4 = p q + pq 4 = p + q = which allows us to consider just one parameter p. Seraphina Lee Spectral Decimation November, 7 9 / 53

30 Spectral Decimation! Three cases for pointwise Laplacian, assuming p + q =, y < z < y V m ( ) 4 m m f (z) = (f (z) f (y ) f (y )) if i(a ) = i(a ) pq ( ) 4 m m f (z) = (f (z) qf (y ) pf (y )) if i(a ) = i(a ) + pq ( ) 4 m m f (z) = (f (z) pf (y ) qf (y )) if i(a ) = i(a ) pq (i is a function that depends on the position of the cell A in the interval) where A has vertices z, y and A has vertices z, y. x y z y x Seraphina Lee Spectral Decimation November, 7 3 / 53

31 Spectral Decimation Continued... By the same process, we obtain the extensions y = q(p(x x ) + x (λ m+ ) ) ( 4q + (λ m+ ) )(λ m+ ) q(x + x ) z = (λ m+ ) 4q y = q(p(x x ) + x (λ m+ ) ) ( 4q + (λ m+ ) )(λ m+ ) Given λ m+ ( q), ( + q), and λ m = (λ m+ 4)(λ m+ ) λ m+ 4pq ( ) m with all eigenvalues scaled by 4 pq Seraphina Lee Spectral Decimation November, 7 3 / 53

32 A corollary Note that p, q enter all equations symmetrically, which leads us to this corollary: Corollary Let λ n,p (m), λ (m) n,q be the nth eigenvalues on level m of the Laplacian with measure parameters p and q, respectively. Then λ (m) n,p = λ (m) n,q mod 4 a for some a m. if n 4a Seraphina Lee Spectral Decimation November, 7 3 / 53

33 Resulting Pattern Eigenvalues(m= p=. q=.9) Eigenvalues(m= p=.9 q=.) Seraphina Lee Spectral Decimation November, 7 33 / 53

34 Eigenfunctions m=5 p=. λ = m=5 p=. λ = m=5 p=. λ 3 = m=5 p=. λ 4 = m=5 p=. λ 5 = m=5 p=. λ 6 = m=5 p=. λ 7 = m=5 p=. λ 8 =4.393 Figure: Eigenfunctions on level m = 5 with lowest 8 eigenvalues with p =. Seraphina Lee Spectral Decimation November, 7 34 / 53

35 Eigenfunctions m=5 p=.9 λ = m=5 p=.9 λ = m=5 p=.9 λ 3 = m=5 p=.9 λ 5 = m=5 p=.9 λ 7 = m=5 p=.9 λ 4 = m=5 p=.9 λ 6 = m=5 p=.9 λ 8 =4.393 Figure: Eigenfunctions on level m = 5 with lowest 8 eigenvalues with p =.9 Seraphina Lee Spectral Decimation November, 7 35 / 53

36 One Parameter Family of Laplacians on SG r µ r r r µ r r µ r r r r µ r r µ r r r r µ r r µ r r r r µ µ r r r r r with new IFS {F jk : F jk = F j F k, j, k =,, } Seraphina Lee Spectral Decimation November, 7 36 / 53

37 One Parameter Family Again, we want to make sure the renormalization factor is constant throughout SG, which makes us end up with only one parameter, r = r r to vary. Seraphina Lee Spectral Decimation November, 7 37 / 53

38 Spectral decimation again... Similarly, ( ) m mf (z) = (4f (z) f (y ) f (y ) f (y 3 ) f (y 4 )) µ r if i(a ) = i(a ) ( ) m mf (z) = ( (µ (f (z) f (y ) f (y )) + µ (f (z) f (y 3 ) f (y 4 )))) µ r µ + µ if i(a ) = i(a ) + ( ) m mf (z) = ( (µ (f (z) f (y ) f (y )) + µ (f (z) f (y 3 ) f (y 4 )))) µ r µ + µ if i(a ) = i(a ) where A has vertices y, y, z and A has vertices y, y 3, z. x y, y, w z w y, z z y, x y, w y, x Seraphina Lee Spectral Decimation November, 7 38 / 53

39 Extension Algorithm w (x, x, x, λ m+, r) = 4x ( + (r + r )(λ m+ 6) + λ m+ ) γ(r, λ m+ ) + (x + x )( 8 + 3λ m+ λ m+ + λ3 m+ ) γ(r, λ m+ ) + (x + x )(r ( + 6λ m+ λ m+ + λ3 m+ )) γ(r, λ m+ ) + (x + x )(r( 3λ λ m+ + λ3 m+ )) γ(r, λ m+ ) x y, y, w w z y, z z y, x x y w, y, Seraphina Lee Spectral Decimation November, 7 39 / 53

40 Extension Algorithm z (x, x, x, λ m+, r) = (x + x )(8 6λ m+ + λ m+ + r ( 8λ m+ + λ m+ ) γ(r, λ m+ ) + 4r(x + x )(4 7λ m+ + λ m+ ) γ(r, λ m+ ) + 4x ( + 5λ m+ + r ( 6 + 9λ m+ 9λ m+ + λ3 m+ )) γ(r, λ m+ ) + 4x ( 9λ m+ + λ3 m+ + r( + 4λ m+ 9λ m+ + λ m+)) γ(r, λ m+ ) x y, y, w w z y, z z y, x x y w, y, Seraphina Lee Spectral Decimation November, 7 4 / 53

41 Extension Algorithm y, (x, x, x, λ m+, r) = x (λ m+ ) (5 8λ m+ + λ m+ ) γ(r, λ m+ ) + r x ( 56λ m+ + 45λ m+ λ3 m+ + λ4 m+ ) γ(r, λ m+ ) + 4rx (8 78λ m+ + 5λ m+ λ3 m+ + λ4 m+ ) γ(r, λ m+ ) + 4rx ( r(λ m+ 6) 3λ m+ ) γ(r, λ m+ ) + 4rx (4 7λ m+ + λ m+ + r(6 7λ m+ + λ m+ )) γ(r, λ m+ ) x y, y, w w z y, z z y, x x y w, y, Seraphina Lee Spectral Decimation November, 7 4 / 53

42 Eigenvalue Extension Algorithm Where γ(r, λ m+ ) = (4 ( + 3r)λ m+ + ( + r)λ m+ )( 3 + 3λ m+ λ m+ + λ3 m+ ) + r (4 ( + 3r)λ m+ + ( + r)λ m+ )( 8 + 7λ m+ λ m+ + λ3 m+ ) + r(4 ( + 3r)λ m+ + ( + r)λ m+ )( 6 + 3λ m+ λ m+ + λ3 m+ ) This equation is quintic in λ m+, but has explicit zeros, referred to as b, b,, b 5 Seraphina Lee Spectral Decimation November, 7 4 / 53

43 Eigenvalue Extension Algorithm λ m(λ m+, r) = λ m+(λ m+ ) (5 8λ m+ + λ m+ ) 4r( 4 + r(λ m+ ) + λ m+ ) + r 3 λ m+ (7 6λ m+ + 67λ m+ 4λ3 m+ + λ4 m+ ) 4r( 4 + r(λ m+ ) + λ m+ ) + r λ m+ (44 37λ m+ + 89λ m+ 4λ3 m+ + 3λ4 m+ ) 4r( 4 + r(λ m+ ) + λ m+ ) + rλ m+(4 33λ m+ + 73λ m+ 38λ3 m+ + 3λ4 m+ ) 4r( 4 + r(λ m+ ) + λ m+ ) This equation is quintic in λ m+ and we can only find numerical solutions. Seraphina Lee Spectral Decimation November, 7 43 / 53

44 Forbidden eigenvalues born on each level Similarly as before, the spectral decimation process depends on λ m+ not being certain values, and by a similar argument, we can characterize all forbidden eigenvalues that are born on each level m to complete the spectrum: λ m = m = m = 3... m b... b... b m +3 b m +3 b m +3 b m 3 b m Seraphina Lee Spectral Decimation November, 7 44 / 53

45 Eigenfunctions m=3 r=3 λ =.536 m=3 r=3 λ = m=3 r=3 λ 3 = m=3 r=3 λ 4 = Figure: Eigenfunctions on level m = 3 with lowest 4 eigenvalues with r = 3 Seraphina Lee Spectral Decimation November, 7 45 / 53

46 Eigenfunctions m=3 r=.5 λ =5.98 m=3 r=.5 λ = m=3 r=.5 λ 3 = m=3 r=.5 λ 4 = Figure: Eigenfunctions on level m = 3 with lowest 4 eigenvalues with r =.5 Seraphina Lee Spectral Decimation November, 7 46 / 53

47 Sturm-Liouville Systems The Sturm-Liouville equation is a second-order homogeneous linear differential equation of the form On the unit interval d dx [ du ] [ ] p(x) + λρ(x) q(x) u = () dx u + λu = () Seraphina Lee Spectral Decimation November, 7 47 / 53

48 Analog of Sturm Liouville Theorem Let λ i be the ith eigenvalue and f i be the eigenfunction for λ i. (a) For any eigenfunction f of the interval, there is exactly one local extremum between two consecutive zeros. (b) f i has i zeros. (c) If λ i < λ j and x k, x k+ are consecutive zeros of f i, then f j has at least one zero in [x k, x k+ ]. Seraphina Lee Spectral Decimation November, 7 48 / 53

49 Zeros Figure: Zeros of eigenfunctions for p =.8 Seraphina Lee Spectral Decimation November, 7 49 / 53

50 Spectral Operator Let λ j, u j be an orthonormal basis of eigenfunctions and their corresponding eigenvalues, with λ j listed in increasing order. Define the spectral operator f ( )u = f (λ j ) u, u j u j j= Seraphina Lee Spectral Decimation November, 7 5 / 53

51 Heat Equation Problem We seek u(x, t) such that and u(x, t) t = x u(x, t) u(x, ) = f (x) Classical solution given by u(x, t) = λ j e λ j t u j (x) u j (y)f (y)dµ(y) Seraphina Lee Spectral Decimation November, 7 5 / 53

52 Wave Equation on the Interval We can also use these methods to solve the wave equation and Classical solution given by u(x, t) = λ j u(x, t) t = x u(x, t) u(x, ) = u u(x, ) = f (x) t sin t λ j u j (x) λj u j (y)f (y)dµ(y) Seraphina Lee Spectral Decimation November, 7 5 / 53

53 References [ABS] Allan, Adam, Michael Barany, and Robert S. Strichartz. 9. Spectral Operators on the Sierpinski Gasket I. Complex Variables and Elliptic Equations 54 (6): 543. doi:.8/ [FKLS] Fang, Sizhen, Dylan A. King, Eun Bi Lee, and Robert S. Strichartz. 7. Spectral Decimation for Families of Self-Similar Symmetric Laplacians on the Sierpinski Gasket. arxiv:79.3 [Str] Strichartz, Robert S. 6. Differential Equations on Fractals: A Tutorial. Princeton, N.J: Princeton University Press. [W] Fang, Sizhen, Dylan A. King, and Eun Bi Lee. Spectral Decimation for Families of Self-Similar Symmetric Laplacians on the Sierpinski Gasket. kingda6/. Seraphina Lee Spectral Decimation November, 7 53 / 53