Analysis on Fractals Jun Kigami Graduate School of Informatics - PowerPoint PPT Presentation

Analysis on Fractals Jun Kigami Graduate School of Informatics Kyoto University Kyoto 606-8501, Japan e-mail:kigami@i.kyoto-u.ac.jp

1 Long Introduction Fractals = Models of natural objects: Coast line, Tree, etc Physical Phenomena on these object ⇒ Analysis on Fractals ↓ ↓ Heat equation: ∂ u Heat ⇒ ∂ t = ∆ u Wave equation: ∂ 2 u Wave ⇒ ∂ t 2 = ∆ u ∂ 2 ∂ 2 ∆ = Laplacian = ∂ x 12 + . . . + ∂ x n 2 on the Euclidean space R n What is the ∆ = Laplacian on a Fractal?

Hata’s tree-like set

The Sierpinski Gasket K dim H K = log 3 log 2 the Hausdor ff dimension with respect to the Euclidean metric

Short History 1987: Construction of the Brownian motion on the Sierpinski gasket– Probability method ( Kusuoka, Goldstein ) 1989: Construction of the Laplacian on the Sierpinski gasket – Analytic method

p 1 1 1 1 p 2 p 3 G 0 G 1 G 2 The Sierpsinski Gasket: Approximation by Graphs G m F i ( z ) = ( z − p i ) / 2 + p i for i = 1 , 2 , 3 V 0 = { p 1 , p 2 , p 3 } V m +1 = F 1 ( V m ) ∪ F 2 ( V m ) ∪ F 3 ( V m ) K = ∪ m ≥ 0 V m : the Sierpinski gasket K = F 1 ( K ) ∪ F 2 ( K ) ∪ F 3 ( K )

x x − h x + h Definition of “Laplacian” ∆ For R , ( ∆ u )( x ) = d 2 u 1 ⇥ ⇤ dx 2 = def lim ( u ( x + h ) − u ( x )) + ( u ( x − h ) − u ( x )) h 2 h → 0

2 − m y x ’s = V m, x : direct neighbors of x in V m x ∈ V m Analogously, for the Sierpinski gasket K , define � H m,x u = ( u ( y ) − u ( x )) : Graph Laplacian y ∈ V m,x Note that h = | y − x | = 2 − m . h 2 = | y − x | 2 = 4 − m , hence m →∞ 4 m H m,x u ( ∆ u )( x ) = lim

{ u | u ∈ C ( K ) , ∆ u = 0 } : infinite dimension, dense in C ( K ) ↑ the collection of continous functions on K No di ff usion on the Sierpinski gasket!! Space-Time Scaling is di ff erent from R n ← time = distance 2 . ( ∆ ν u )( x ) = lim m →∞ 5 m H m,x u 5... Why? What is ν ?

Harmonic functions On R n , � | y − x | = r u ( y ) dS ∆ u = 0 ⇔ u ( x ) = : Average over a sphere � | y − x | = r dS On the Sierpinski gasket, u is harmonic ⇔ def H m,x u = 0 for any x ∈ V m \ V 0 and m ≥ 1. V 0 : the boundary

f ( p 1 ) = a 2 a + 2 b + c 2 a + b + 2 c 5 5 f ( p 2 ) = b a + 2 b + 2 c f ( p 3 ) = c 5   a To get a harmonic function f with boundary value b   c

Iteration of the procedure gives     a a  =  . b b h V 0 def Harmonic function f with boundary value   c c       1 0 0  , ψ 2 =  , ψ 3 = ψ 1 = def h V 0 0 def h V 0 1 def h V 0 0     0 0 1

Definition of µ -Laplacian ∆ µ u µ : a Borel regular probability measure on K — a mass distribution on K For x ∈ V ∗ = ∪ m ≥ 0 V m 1 ⇥ 5 ⇤ m ( ∆ µ u )( x ) = def lim H m,x u ✏ 3 m →∞ ψ m x dµ K

0 0 1 2 2 1 5 5 5 5 2 2 0 1 0 5 5 ∈ V m ∈ V m +1 ψ m x : piecewise harmonic function with � 1 if p = x ψ m x ( p ) = otherwise on V m 0 ✏ ψ m x dµ : local scaling of mass K

p 1 1 1 1 1 1 1 1 1 1 1 p 2 p 3 1 1 1 3 5 = Resistance scaling Attach a resistor of resistance 1 to each edge of V m for any m . Then The e ff ective resistance between p 1 and p 2 = 3 ⇥ 5 ⇤ m 2 3

1 9 1 1 1 3 9 9 1 1 1 9 9 1 1 1 1 1 1 3 3 9 9 9 9 def the normalized log 3 ν = log 2-dim. Hausdor ff measure ⇥ 1 3 , 1 3 , 1 ⇤ = the self-similar measure with weight 3 Then ✏ ⇥ 1 ⇤ m +1 1 ⇥ 5 H m,x u = 3 ⇤ m ψ m 25 m H m,x u x d ν = 2 ⇒ � 3 K ψ m 3 x d ν K

First way to Laplacian: Fuctional Analysis The standard resistance form on K : ( E , F ) F = { u | lim m →∞ E m ( u, u ) < + ∞ } E ( u, v ) = lim m →∞ E m ( u, v ) ← Energy E m ( u, u ) = 1 ⇥ 5 ⇤ m ( u ( p ) − u ( q )) 2 . � where 2 3 ( p, q ) is an edge of the Graph G m Fact: E m ( u, u ) ≤ E m +1 ( u, u ) Theorem 1. F ⊆ C ( K ) . ( E , F ) is a local regular Dirichlet form on L 2 ( K, µ ) . In particular, ( E , F ) is closed and ✏ E ( u, v ) = − u ∆ µ vdµ. K local regular Dirichlet form ↔ di ff usion process closed form ↔ Laplacian = self-adjoint operator

n ✏ ∂ u ∂ v � Dirichlet form ( E , F ) on L 2 ( X, µ ) E ( u, v ) = dx ∂ x i ∂ x i R n i =1 F = H 1 ( R ): Sobolev space E : non-negative quadratic form with the Markov property ↓ ↓ � � E ( u, v ) = X u ( Lv ) dµ E ( u, v ) = R n u ( − ∆ v ) dx n ∂ 2 � − L : Laplacian, L ≥ 0, self-adjoint ∆ = ∂ x i 2 i =1 ↓ ↓ ∂ u ∂ u ∂ t = − Lu : Heat equation ∂ t = ∆ u ↓ ↓ u ( x, t ) = e − tL u 0 = initial condition u ( x, t ) = e t ∆ u 0 ↓ ↓ Process ( { X t } t> 0 , { P x } x ∈ X ) with The Brownian motion on R n E x ( u ( X t )) = ( e − tL u )( x )

Eigenvalue distribution of − ∆ ν u Fact : − ∆ ν has compact resolvent. ⇒ Spectrum = eingenvalues the sequence of eingenvalues taking the multiplicity into account 0 < λ 1 ≤ λ 2 ≤ λ 3 ≤ . . . Theorem 2 (Fukushima-Shima, Kigami-Lapidus) . . Let N ( x ) = def # { i | λ i ≤ x } : the eigenvalue counting function . Then N ( x ) = G (log x ) x d s / 2 + O (1) as x → ∞ , where G ( · ) : log 5 -periodic and discontinuous with 0 < inf G ( x ) < sup G ( x ) < + ∞ and d S = log 9 log 5 : spectral dimension of the SG

Comparison with the Euclidean case Ω : a bounded domain in R n N ( x ) = C n | Ω | n x n/ 2 + o ( x n/ 2 ) Weyl where | Ω | n : n -dim volume of Ω and C n : a constant only depends on n . (1) d S ⌘ = the Hausdor ff dim. log 3 / log 2 x →∞ N ( x ) /x d S / 2 does not exists. (2) lim

Asymptotic behavior of the heat kernel Heat kernel p µ ( t, x, y ) = the fundamental solution of the Heat equation: ✏ ∂ u ∂ t = ∆ µ u � u ( t, x ) = p µ ( t, x, y ) u 0 ( y ) dµ K Theorem 3 (Barlow-Perkins) . For 0 < t ⌃ 1 , ⌃ ⇧ 1 / ( d w − 1) ⌥ ⌅ | x � y | d w c 1 p ν ( t, x, y ) ⇥ t d S / 2 exp � c 2 , t where d w = log 5 log 2 : the walk dimension . sub-Gaussian heat kernel estimate d w > 2: slower than the Gaussian

On R n or a complete Riemannian manifold with non-negative Ricci curvature, we have the following Gaussian estimate ⌃ ⌥ | x � y | 2 c 1 p ( t, x, y ) ⇥ t ) exp : � c 2 Li-Yau , ⌫ V ( x, t where V ( x, r ) is the (Riemannian) volume of a ball of radius r . d w = 2 — Gaussian

(metrizable topological) Space + Structure ⇒ “Geometry” Example: Riemannian structure, i.e. inner products on tangent spaces ⇒ Riemannian metric Energy form (Resistance form) on the Sierpinski gasket ⇒ Resistance metric Energy forms/Dirichlet forms ( ⊇ Resistance forms): one of structures

How can we construct a energy/Dirichlet form on a set? One possible way: Discrete approximation X : a metrizable topological space, For simplicity, assume that X is compact V 0 ⊆ V 1 ⊆ V 3 ⊆ . . . ⊆ X , V m : a finite set Assume ⇣ X = the closure of V ∗ = V m . m ≥ 0 Prepare E m = Energy/Dirichlet form on V m — What is this?

Energy form on a finite set V = resistance network ( V, ⇤ ): a resistance network � def ✓ x ⌘ = y ⇣ V , ⇤ ( x, y ) ⇣ (0 , ✏ ]: resistance between x and y non-negative, symmetric : ⇤ ( x, y ) = ⇤ ( y, x ) ⌥ 0 connected : ✓ x ⌘ = y, ◆ x 1 , . . . , x n , x 1 = x, x n = y and ✓ i, ⇤ ( x i , x i +1 ) > 0 . For u, v ⇣ ⌃ ( V ) = R V , define 1 1 � E ( V, ρ ) ( u, v ) = ⇤ ( x, y )( u ( x ) � u ( y ))( v ( x ) � v ( y )) 2 def x,y ⌥ V,x � = y E ( V, ρ ) : energy associated with a resistance network ( V, ⇤ )

Basic property: E ( V, ρ ( u, u ) ≥ 0 and E ( V, ρ ) ( u, u ) = 0 ⇔ u ≡ a constant Resiscance associted with ( V, ⇤ ): A, B ⊆ V, A ∩ B = ∅ 1 � ✓ ◆ � u ∈ ⌃ ( V ) , u | A = 0 , u | B = 1 R ( V, ρ ) ( A, B ) = def sup � E ( V, ρ ) ( u, u ) R ( V, ρ ) ( A, B ): e ff ective resistance between A and B .

X : a compact metrizable space, ( V m , ρ m ): weighed graphs, V m ⊆ V m +1 ⊆ X = the closure of V ∗ . When does E ( V m , ρ m ) converge ? “ converge ” ⇔ def (1) and (2) (1) ∃ a resonably large collection of functions F on X such that ∀ u ∈ F , m →∞ E ( V m , ρ m ) ( u | V m , u | V m ) = lim def E ( u, u ) exists and nonnegative. (2) If A, B ⊆ X, A ∩ B = ∅ , A, B : resonably large sets m →∞ R ( V m , ρ m ) ( A ∩ V m , B ∩ V m ) = R ( A, B ) lim exists and nonnegative.

( E , F ): energy on X , R ( A, B ): e ff ective resistance between A and B Again we ask When does { ( V m , ρ m ) } m ≥ 0 converge? and if it converges, then What does ( E , F ) do? Hopefully, ( E , F ) is a regular local Dirichlet form, which produces di ff usions and/or Laplacian.

Examples of convergence : ✓ k � � k = 0 , 1 , . . . , 2 m ◆ (1) the unit interval [0 , 1]: V m = � 2 m ⇥ k � 1 if | k − l | = 1, 2 m , l ⇤ 2 m = ρ m 2 m if | k − l | ≥ 2. ∞ ∀ u : [0 , 1] → R , E ( V m , ρ m ) ( u | V m , u | V m ) ≤ E ( V m +1 , ρ m +1 ) ( u | V m +1 , u | V m +1 ) ✏ 1 ( u ⇧ ( x )) 2 dx → E ( u, u ) = as m → ∞ 0 ( E , F ) is a local regular Dirichlet from on L 2 ([0 , 1] , dx ) → d 2 the Brownian motion on [0 , 1] and the Laplacian dx 2