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 ⇒ Heat equation: ∂u ∂t = ∆u Wave ⇒ Wave equation: ∂2u ∂t2 = ∆u ∆ = Laplacian = ∂2 ∂x12 + . . . + ∂2 ∂xn2 on the Euclidean space Rn What is the ∆ = Laplacian on a Fractal?

Hata’s tree-like set

The Sierpinski Gasket K dimH K = log 3 log 2 the Hausdorff 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

1 1 1 G0 G1 G2

p1 p2 p3 The Sierpsinski Gasket: Approximation by Graphs Gm Fi(z) = (z − pi)/2 + pi for i = 1, 2, 3 V0 = {p1, p2, p3} Vm+1 = F1(Vm) ∪ F2(Vm) ∪ F3(Vm) K = ∪m≥0Vm: the Sierpinski gasket K = F1(K) ∪ F2(K) ∪ F3(K)

x − h x x + h

Definition of “Laplacian” ∆ For R, (∆u)(x) = d2u dx2 =

def lim h→0

1 h2 ⇥ (u(x + h) − u(x)) + (u(x − h) − u(x)) ⇤

x y

2−m ’s = Vm, x: direct neighbors of x in Vm x ∈ Vm Analogously, for the Sierpinski gasket K, define Hm,xu =

• y∈Vm,x

(u(y) − u(x)) : Graph Laplacian Note that h = |y − x| = 2−m. h2 = |y − x|2 = 4−m, hence (∆u)(x) = lim

m→∞ 4mHm,xu

{u|u ∈C(K), ∆u = 0} : infinite dimension, dense in C(K) ↑ the collection of continous functions on K No diffusion on the Sierpinski gasket!! Space-Time Scaling is different from Rn ← time = distance2. (∆νu)(x) = lim

m→∞ 5mHm,xu

5... Why? What is ν?

Harmonic functions

On Rn, ∆u = 0 ⇔ u(x) =

• |y−x|=r u(y)dS
• |y−x|=r dS

: Average over a sphere On the Sierpinski gasket, u is harmonic ⇔

def Hm,xu = 0 for any x ∈ Vm\V0 and m ≥ 1.

V0: the boundary

f(p1) = a f(p2) = b f(p3) = c a + 2b + 2c 5 2a + 2b + c 5 2a + b + 2c 5

To get a harmonic function f with boundary value   a b c  

Iteration of the procedure gives hV0   a b c   =

def Harmonic function f with boundary value

  a b c   . ψ1 =

def hV0

  1   , ψ2 =

def hV0

  1   , ψ3 =

def hV0

  1  

Definition of µ-Laplacian ∆µu µ: a Borel regular probability measure on K — a mass distribution on K For x ∈ V∗ = ∪m≥0Vm (∆µu)(x) =

def lim m→∞

1 ✏

K

ψm

x dµ

⇥5 3 ⇤m Hm,xu

1

1 5 2 5 2 5 1 5 2 5 2 5

∈ Vm+1 ∈ Vm ψm

x : piecewise harmonic function with

ψm

x (p) =

• 1

if p = x

• therwise on Vm

✏

K

ψm

x dµ: local scaling of mass

1 1 1 1 1 1 1 1 1 1 1 1 1

p1 p2 p3 3 5 = Resistance scaling Attach a resistor of resistance 1 to each edge of Vm for any m. Then The effective resistance between p1 and p2 = 3 2 ⇥5 3 ⇤m

1

1 3 1 3 1 3

1 9 1 9 1 9 1 9 1 9 1 9 1 9 1 9 1 9

ν =

def the normalized log 3

log 2-dim. Hausdorff measure = the self-similar measure with weight ⇥1 3, 1 3, 1 3 ⇤ Then ✏

K

ψm

x dν = 2

⇥1 3 ⇤m+1 ⇒ 1

• K ψm

x dν

⇥5 3 ⇤m Hm,xu = 3 25mHm,xu

First way to Laplacian: Fuctional Analysis The standard resistance form on K: (E, F) F = {u| lim

m→∞ Em(u, u) < +∞}

E(u, v) = lim

m→∞ Em(u, v) ← Energy

where Em(u, u) = 1 2

• (p, q) is an edge of the Graph Gm

⇥5 3 ⇤m (u(p) − u(q))2 . Fact: Em(u, u) ≤ Em+1(u, u) Theorem 1. F ⊆ C(K). (E, F) is a local regular Dirichlet form on L2(K, µ). In particular, (E, F) is closed and E(u, v) = − ✏

K

u∆µvdµ. local regular Dirichlet form ↔ diffusion process closed form ↔ Laplacian = self-adjoint operator

Dirichlet form (E, F) on L2(X, µ) E(u, v) = ✏

Rn n

• i=1

∂u ∂xi ∂v ∂xi dx E : non-negative quadratic form F = H1(R): Sobolev space with the Markov property ↓ ↓ E(u, v) =

• X u(Lv)dµ

E(u, v) =

• Rn u(−∆v)dx

−L: Laplacian, L ≥ 0, self-adjoint ∆ =

n

• i=1

∂2 ∂xi2 ↓ ↓ ∂u ∂t = −Lu : Heat equation ∂u ∂t = ∆u ↓ ↓ u(x, t) = e−tLu0= initial condition u(x, t) = et∆u0 ↓ ↓ Process ({Xt}t>0, {Px}x∈X) with The Brownian motion on Rn Ex(u(Xt)) = (e−tLu)(x)

Eigenvalue distribution of −∆νu Fact: −∆ν has compact resolvent. ⇒ Spectrum = eingenvalues the sequence of eingenvalues taking the multiplicity into account 0 < λ1 ≤ λ2 ≤ λ3 ≤ . . . Theorem 2 (Fukushima-Shima, Kigami-Lapidus). . Let N(x) =

def #{i|λi ≤ x} :the eigenvalue counting function. Then

N(x) = G(log x)xds/2 + O(1) as x → ∞, where G(·): log 5-periodic and discontinuous with 0 < inf G(x) < sup G(x) < +∞ and dS = log 9 log 5 : spectral dimension of the SG

Comparison with the Euclidean case Ω: a bounded domain in Rn N(x) = Cn|Ω|nxn/2 + o(xn/2) Weyl where |Ω|n: n-dim volume of Ω and Cn: a constant only depends on n. (1) dS ⌘= the Hausdorff dim. log 3/ log 2 (2) lim

x→∞ N(x)/xdS/2 does not exists.

Asymptotic behavior of the heat kernel Heat kernel pµ(t, x, y) = the fundamental solution of the Heat equation: ∂u ∂t = ∆µu u(t, x) = ✏

K

pµ(t, x, y)u0(y)dµ Theorem 3 (Barlow-Perkins). For 0 < t ⌃ 1, pν(t, x, y) ⇥ c1 tdS/2 exp ⌃ c2 ⌅|x y|dw t ⇧1/(dw−1)⌥ , where dw = log 5 log 2: the walk dimension. sub-Gaussian heat kernel estimate dw > 2: slower than the Gaussian

On Rn or a complete Riemannian manifold with non-negative Ricci curvature, we have the following Gaussian estimate p(t, x, y) ⇥ c1 V (x, ⌫ t) exp ⌃ c2 |x y|2 t ⌥ : Li-Yau, where V (x, r) is the (Riemannian) volume of a ball of radius r. dw = 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 V0 ⊆ V1 ⊆ V3 ⊆ . . . ⊆ X, Vm: a finite set Assume X = the closure of V∗ = ⇣

m≥0

Vm. Prepare Em = Energy/Dirichlet form on Vm— 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, ◆x1, . . . , xn, x1 = x, xn = y and ✓i, ⇤(xi, xi+1) > 0. For u, v ⇣ ⌃(V ) = RV , define E(V,ρ)(u, v) =

def

1 2

• x,y⌥V,x=y

1 ⇤(x, y)(u(x) u(y))(v(x) v(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 = ∅ R(V,ρ)(A, B) =

def sup

✓ 1 E(V,ρ)(u, u)

• u ∈ ⌃(V ), u|A = 0, u|B = 1

◆ R(V,ρ)(A, B): effective resistance between A and B.

X: a compact metrizable space, (Vm, ρm): weighed graphs, Vm ⊆ Vm+1 ⊆ X = the closure of V∗. When does E(Vm,ρm) converge? “converge” ⇔

def (1) and (2)

(1) ∃ a resonably large collection of functions F on X such that ∀u ∈ F, lim

m→∞ E(Vm,ρm)(u|Vm, u|Vm) = def E(u, u)

exists and nonnegative. (2) If A, B ⊆ X, A ∩ B = ∅, A, B : resonably large sets lim

m→∞ R(Vm,ρm)(A ∩ Vm, B ∩ Vm) = R(A, B)

exists and nonnegative.

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

Examples of convergence: (1) the unit interval [0, 1]: Vm = ✓ k 2m

• k = 0, 1, . . . , 2m◆

ρm ⇥ k 2m, l 2m ⇤ =

• 1

2m

if |k − l| = 1, ∞ if |k − l| ≥ 2. ∀u : [0, 1] → R, E(Vm,ρm)(u|Vm, u|Vm) ≤ E(Vm+1,ρm+1)(u|Vm+1, u|Vm+1) → E(u, u) = ✏ 1 (u⇧(x))2dx as m → ∞ (E, F) is a local regular Dirichlet from on L2([0, 1], dx) → the Brownian motion on [0, 1] and the Laplacian

d2 dx2

(2) the Sierpinski gasket K: F = {u| lim

m→∞ Em(u, u) < +∞}

E(u, v) = lim

m→∞ Em(u, v) ← Energy

where Em(u, u) = 1 2

• (p, q) is an edge of the Graph Gm

⇥5 3 ⇤m (u(p) − u(q))2 . Fact: Em(u, u) ≤ Em+1(u, u)

(3) the unit square [0, 1]2: Vm = ✓⇥ k 2m, l 2m ⇤

• k, l ∈ {0, . . . , 2m}

◆ . ρm ⇥⇥ k1 2m, l1 2m ⇤ , ⇥ k2 2m, l2 2m ⇤⇤ =

• 1

if |k1 − k2| + |l1 − l2| = 1, ∞

• therwise.

E(Vm,ρm)(u|Vm, u|Vm) ≤ E(Vm+1,ρm+1)(u|Vm+1, u|Vm+1) does not hold!!, but ∀u ∈ C2, E(Vm,ρm)(u|Vm, u|Vm) → ✏

[0,1]2

⇥∂2u ∂x2 + ∂2u ∂y2 ⇤ dxdy as m → ∞ (E, F): a local regular Dirichlet from producing the Brownian motion and the Laplacian

(4) the Sierpinski carpet K: S = {1, 2, . . . , 8}. Fi : R2 → R2 Fi(x) =

def

1 2(x − pi) + pi K = ⇣

i∈S

Fi(K) Define V0 =

def {p1, p3, p5, p7} and Vm+1 = def ∪i∈SFi(Vm).

the Sierpinski carpet

To make {(Vm, ρM)}m≥0 converge, one should do renormalization, i. e. Rm =

def R(Vm,ρm)(A ∩ Vm, B ∩ Vm),

⌘ ρm =

def ρm/Rm.

Then R(Vm,⌘

ρm)(A ∩ Vm, B ∩ Vm) = 1

E(Vm,⌘

ρm) converges as m → ∞

Very hard to prove. Due to Barlow-Bass, Kusuoka-Zhou. → the Brownian motion on the Sierpinski carpet

Energy form on a finite set V = resistance network (V, ρ): a resistance network ⇔

def

∀x ̸= y ∈ V , ρ(x, y) ∈ (0, ∞]: resistance between x and y non-negative, symmetric : ρ(x, y) = ρ(y, x) ≥ 0 connected : ∀x ̸= y, ∃x1, . . . , xn, x1 = x, xn = y and ∀i, ρ(xi, xi+1) > 0. E(V,ρ)(u, v) =

def

1 2

• x,y∈V,x̸=y

1 ρ(x, y)(u(x) − u(y))(v(x) − v(y)) Easy case: monotonicity, i.e. E(Vm,ρm)(u|Vm, u|Vm) ≤ E(Vm+1,ρm+1)(u|Vm+1, u|Vm+1) , ⇒ Resisnance form ⇒ Harmonic function, Green function, Dirichlet form, heat kernel, etc 35

2 Resistance forms

Definition 1. X: a set. (E, F): resistance form on X ⇔

def (RF1) through (RF5) hold.

(RF1) F: a linear subspace of ℓ(X), 1 ∈ F, E : F × F → R, non-negative symmetric E(u, u) = 0 ⇔ u is constant on X. (RF2) u ∼ v⇔

def u − v ≡ a constant on X. Then (F/∼, E) is a Hilbert space.

(RF3) x ̸= y ⇒ ∃u ∈ F such that u(x) ̸= u(y). 36

(RF4) ∀x, y ∈ X, R(E,F)(x, y) = sup |u(x) − u(y)|2 E(u, u) : u ∈ F, E(u, u) > 0

• < +∞

(RF5) Markov property: Define u by u(p) =

def

⎧ ⎪ ⎨ ⎪ ⎩ 1 if u(p) ≥ 1, u(p) if 0 < u(p) < 1, if u(p) ≤ 0. Then ∀u ∈ F, u ∈ F and E(¯ u, ¯ u) ≤ E(u, u). 37

R(E,F)(x, y): the resistance metric on X associated with (E, F) Theorem 4. R(E,F)(·, ·) is a meric on X. ∀u ∈ F, x, y ∈ X, |u(x) − u(y)|2 ≤ R(E,F)(x, y)E(u, u). For simplicity, we use R(x, y) instead of R(E,F)(x, y). 38

Examples of resistance forms

(1) 1-dim. Brownian motion: E(u, v) =

• R

du dx dv dxdx F = {u|E(u, u) < +∞} = H1(R) R(x, y) = |x − y| 39

(2) Random walks on (weighted) graphs (V, C): V : a countable set, (V, ρ): a resisitance network ⇒ C(x, y) =

def 1/ρ(x, y): conductance

C =

def (C(x, y))x,y∈V,x̸=y

(V, C) : weighted graph on V 40

Random walk associated with (V, C): C(x) =

• y

C(x, y): the weight of x P(x, y) = C(x, y) C(x) : the transition probability from x to y P n(x, y) =

• z∈V

P n−1(x, z)P(z, y): the transition probability at the time n P n(x, y): the “heat kernel” associated with the random walk 41

Resistance form associated with (V, C): F = {u|u : V → R,

• x,y

C(x, y)(u(x) − u(y))2 < +∞} E(u, v) = 1 2

• x,y

C(x, y)(u(x) − u(y))(v(x) − v(y)). (E, F) is a resistance form on V . 42

(3) α-stable process on R1: α ∈ (1, 2] E(α)(u, u) =

• R
• R

|u(x) − u(y)|2 |x − y|1+α dxdy =

• R

u(x)

• (−∆)α/2u
• (x)dx

F(α) = {u|u ∈ C(R), E(α)(u, u) < +∞} R(α)(x, y) = c|x − y|α−1 for α ∈ (1, 2). For α = 2, it corresponds to the Brownian motion on R1. Laplacian L = −(−∆)α/2: not a local operator p(t, x, y) ≈ min

• t−1/α,

t |x − y|1+α

• 43

2.1 Resistance forms on a finite set

V : a finite set, ℓ(V ) =

def RV

Definition 2. H : ℓ(V ) → ℓ(V ): a Laplacian on V ⇔

def H = (Hpq)p,q∈V ,

(L1) H is non-positive definite, (L2) Hu = 0 ⇔ u is a constant on V , (L3) ∀p ̸= q ∈ V , Hpq ≥ 0. H: a Laplacian ⇔ ρ(x, y) = 1/Hxy: resistance network Proposition 1. E: a resistance form on V ⇔ ∃a Laplacian H such that E(u, u) = −tuHu = 1 2

• p,q∈V

Hpq(u(p) − u(q))2 ⇒ (Write E = EH) 44

Definition 3 (Monotonicity). V1, V2: finite sets, V1 ⊆ V2, Hi: a Laplacian on Vi (V1, H1) ≤ (V2, H2) ⇔

def ∀u ∈ ℓ(V1),

EH1(u, u) = inf

v∈ℓ(V2),v|V1=u EH2(v, v)

⇒ ∀u ∈ ℓ(V2) EH1(u|V1, u|V1) ≤ EH2(u, u) 45

Proposition 2. Ei = EHi, Hi: a Laplacian on Vi, Let T : ℓ(V1) → ℓ(V1), J : ℓ(V1) → ℓ(V2\V1) and X : ℓ(V2\V1) → ℓ(V2\V1) given by H2 = T

tJ

J X

• .

Then [H2]V1 =

def T − tJX−1J is a Laplacian on V1

and (V1, H1) ≤ (V2, H2) ⇔ H1 = [H2]V1. 46

V : a finite set, H: a Laplacian on V ⇒ RH =

def the associated resistance metric on V

RH(x, y): effective resistance between x and y Facts: (1) V1, V2: finite sets, Hi: a Laplacian on Vi. Then R[H2]V1 = RH2|V1×V1 (2) RH determines H, i.e. H, H′: Laplacians on V . H = H′ ⇔ RH = RH′. Corollary 5. Hi: a Laplacian on Vi, V1 ⊆ V2, then RH1 = RH2|V1×V1 ⇔ (V1, H1) ≤ (V2, H2). 47

V0 ⊆ V1 ⊆ V2 ⊆ · · · : an increasing sequence of finite sets, V∗ =

def ∪m≥0Vm

Hi: a Laplacian on Vi, {(Vi, Hi)}i≥0: a compatible sequence ⇔

def ∀i, (Vi, Hi) ≤ (Vi+1, Hi+1).

⇓ ∀u ∈ ℓ(V∗), EH0(u|V0, u|V0) ≤ EH1(u|V1, u|V1) ≤ EH2(u|V2, u|V2) ≤ . . . ⇓ F =

def {u|u ∈ ℓ(V∗), limm→∞ EHm(u|Vm, u|Vm) < +∞}

∀u, v ∈ F, E(u, v) =

def limm→∞ EHm(u|Vm, v|Vm)

Proposition 3. (E, F) is a resistance form on V∗. 48

Problem: V∗ is merely a countable set!! Theorem 6. X =

def the completion of V∗ with repsect to R. Then

u ∈ F can be extended to a continuous function on X. (E, F) can be reagrded as a resistance form on X. 49

3 Geometry of self-similar sets

Standard notations: S: a finite set Wm = Sm = {w1w2 · · · wm|∀i ∈ 1, . . . , m, wi ∈ S} : words of length m W∗ = ∪m≥0Wm Σ = SN = {w1w2 . . . |∀i, wi ∈ S} : the Cantor set Define σ : Σ → Σ: the shift map and σs : Σ → Σ by σ(ω1ω2ω3 . . .) = ω2ω3ω4 . . . σs(ω1ω2ω3 . . .) = sω1ω3 . . . {σs}s∈S: the branched of the inverse of σ Σ =

• s∈S

σs(Σ) : the Cantor set as a self-similar set 50

3.1 Self-similar sets

(X, d): a complete metric space, S: a finite set, Fs : X → S: contractions Facts (1) ∃ unique non-empty compact set K ⊆ X such that K =

• s∈S

Fs(K) (2) Fw1w2···wm =

def Fw1 ◦ Fw2 · · · ◦ Fwm and Kw1w2···wm = def Fw1w2···wm(K).

∀ω = ω1ω2 . . . ∈ Σ, ∀m, Kω1ω2···ωm ⊇ Kω1ω2···ωmωm+1

• m≥0

Kω1ω2···ωm = a single point =

def π(ω)

π : Σ → K: continuous, surjective, ∀s ∈ S, Fs ◦ π = π ◦ σs 51

Eample: the Sierpinski gasket

p1 = π(˙ 1) p2 = π(˙ 2) p3 = π(˙ 3) q1 q3 q2 q1 = π(2˙ 3) = π(3˙ 2) q2 = π(1˙ 3) = π(3˙ 2) q3 = π(1˙ 2) = π(2˙ 1)

F1 F2 F3

52

Example: Hata’s tree-like set

X = C, α ∈ C, |α|, |1 − α| ∈ (0, 1). F1(z) = αz, F2(z) = (1 − |α|2)z + |α|2 0 = π(˙ 1) 1 = π(˙ 2) α = π(1˙ 2) |α|2 = π(11˙ 2) = π(2˙ 1) π(21˙ 2)

F1 F2

53

π determines the topological structure of K, i.e. K = Σ/π Examples: the Sierpinski gasket, Hata tree, the Sierpinski carpet Definition 4 (Self-similar structure). (K, d): a compact metric space, S: a finite set, Fs : K → K: continuous, injective (K, S, {Fs}s∈S): a self-similar structure ⇔

def

∃π : Σ → K, continuous, surjective ∀s ∈ S, Fs◦π = π◦σs. Fs may not be a contraction. 54

What is the boundary of a self-similar structure. C =

def ∪i̸=j∈SKi ∩ Kj

C =

def π−1(C) : critical set

P =

def ∪k≥1σk(C) : post critical set

V0 =

def π(P)

Fact: ∀w ̸= v ∈ Wm, Kw ∩ Kv = Fw(V0) ∩ Fv(V0) ⇒ V0: “bounary” 55

A self-similar structure is called post critically finite (or p.c.f. for short) ⇔

def P is a finite set.

Define {Vm}m=0,1,2,... inductively by Vm+1 =

def ∪i∈SFi(Vm).

Fact: V0 ⊆ V1 ⊆ V2 ⊆ V3 ⊆ . . .: a natural discrete approximation of K 56

(K, S, {Fi}i∈S): a p.c.f. self-similar set Construction of a self-similar sequence of resistance networks: Starting point: (A) D = (Dpq)p,q∈V0: a Laplacian on V0 → 1/Dpq: resistance between p and q (B) (ri)i∈S: a resistance scaling ratio ⇒ ⎧ ⎨ ⎩ E0(u, v) =

def ED(u, v)

Em+1(u, v) =

def

• i∈S

1 riEm(u ◦ Fi, v ◦ Fi)

Then Em = EDm, where Dm is a Laplacian on Vm 57

If (V0, D0) ≤ (V1, D1)

• r equivalently

[D1]V0 = D0, then ∀m, (Vm, Dm) ≤ (Vm+1, Dm+1)

• r equivalently

EDm(u|Vm, u|Vm) ≤ EDm+1(u|Vm+1, u|Vm+1). The the general machinary works and we obtain a resistance form (E, F). 58

Fix the resistance scaling ratio r = (ri)i∈S. Then D0 → D1 → [D1]V0. Define Rr(D0) = [D1]V0, where R : {Laplacian on V0} → {Laplacian on V0}. Finding a self-similar compatible sequence ⇕ Fixed point problem of non-linear operator Rr: Rr(D0) = D0 Nested fractals(Highly symmetric fractals): Lindstrøm(Existence), Sabot, Metz(Uniqueness) General case: Metz, Peiorne Stability 59

4 Analysis by resistance forms

4.1 Regularity of resistance form

Notation. C(X) = {u|u is continuous with respect to R-topology} C0(X) = {u|u ∈ C(X), supp(u) is R-compact.} Definition 5. (E, F) is regular ⇔

def

F ∩ C0(X) is dense in C0(X) in the sense of ||u||∞ = supx∈X |u(x)|. Theorem 7. If (X, R) is compact, then (E, F) is regular. 60

4.2 Green’s function

F(B) =

def {u|u ∈ F, u|B ≡ 0}.

Theorem 8. B ⊆ X: closed – our “boundary” ∃ Unique gB : X × X → [0, +∞) with (GF) gx

B(y) = def gB(x, y) ⇒ gx B ∈ F(B). ∀u ∈ F(B), ∀x ∈ X,

E(gx

B, u) = u(x)

gB(x, y): the Green function with the boundary B

• r the B-Green function

gB(x, x) ≥ gB(x, y) ≥ 0 gB(x, y) = gB(y, x) gB(x, x) > 0 ⇔ x / ∈ B |gB(x, y) − gB(x, z)| ≤ R(y, z) 61

µ: a Borel regular measure ∆µf = u on Bc, f|B = 0 ⇕ f = GB,µu, where (GB,µu)(x) =

• X

gB(x, y)u(y)µ(dy) Green operator GB,µ = the “inverse” of Laplacian ∆µ 62

Moreover, define FB = F(B) + R = {u|u ∈ F, u is constant on B} XB = (X\B) ∪ {B} : shrinking B into a point Then (E, FB) is a resistance form on XB. RB(·, ·): associated resistance metric on XB. Then, gB(x, y) = RB(x, B) + RB(y, B) − RB(x, y) 2 ↑ Gromov product of the metric RB If B = {z}, then RB(x, y) = R(x, y). 63

4.3 Harmonic functions

B ⊆ X: closed– our “boundary” Define F|B = {u|B : u ∈ F} : boundary values Proposition 4. ∀ϕ ∈ F|B, ∃ unique f ∈ F such that f|B = ϕ and E(f, f) = min

u∈F,u|B=ϕ E(u, u)

f: the harmonic function with boundary value ϕ on the boundary B

• r the B-harmonic function with boundary value ϕ.

64

4.4 Dirichlet form associated with (E, F)

Assume that µ: a Radon measure on (X, R) 0 < µ(BR(x, r)) < +∞ for any x ∈ X and any r > 0. Define E1(u, v) = E(u, v) +

• X

uvdµ D = E1-closure of F ∩ C0(X). 65

Theorem 9. (E, F): regular ⇒ (E, D): a regular Dirichlet form on L2(X, µ) ∃“Laplacian” ∆µ: non-positive self-adjoint opeartor on L2(K, µ) such that E(u, v) = −

• K

u∆µvdµ a regular Dirichlet form ↓ a Hunt process, i.e. a strong Markov process with right continuous pathes local ⇒ pathes are continuous. (Diffusion): local or not? in general 66

4.5 Transition density/Heat kernel

µ: a Radon measure on (X, R), 0 < µ(BR(x, r)) < +∞ (E, F): a regular resistance form on X. ↓ (E, D): a regular Dirichlet form on L2(X, µ) ↓ ({Xt}t>0, {Px}x∈X): a Hunt process on X 67

Theorem 10. Assume ∀x, ∈ X, ∀r > 0 BR(x, r): compact ⇒ ∃ p(t, x, y) : (0, ∞) × X × X → [0, ∞), continuous with (TD1) pt,x ∈ D, where pt,x(y) = p(t, x, y). (TD2) p(t, x, y) = p(t, y, x) (TD3) u(t, x) =

• X

p(t, x, y)u(y)µ(dy) ⇔ ∂u ∂t = ∆µu, u(0, x) = u(x) (TD4) p(t + s, x, y) =

• X

p(t, x, z)p(s, z, y)µ(dz) p(t, x, y): the transition density/heat kernel 68

5 Back to p.c.f self-similar sets

Given

• (K, S, {Fi}i∈S) : a p.c.f. self-similar strucure

r = (ri)i∈S : a resistance scaling ratio, Assume ∃D0: a Laplacian on V0 such that Rr(D0) = D0 ⇓ {(Vm, Dm)}m≥0: compatibel sequence ⇓ (E, F): a resistance form on V∗, and the associated resistance metric R the R-closure of V∗ = K ⇔ ∀i, ri < 1 69

the structure (E, F) on K ⇒ R giving a geometry of K Theorem 11. Assume ∀i ∈ S, ri < 1. Then the Hausdorff dimension of (K, R) is given by the unique solution α of

• i∈S

(ri)α = 1. 70

µ: a Borel regular probability measure on K such that O : open ̸= ∅ ⇒ µ(O) > 0 ⇓ (E, F): a regular local Dirichlet form on L2(K, µ) ∆µ: Laplacian, pµ(t, x, y): heat kernel, i.e. pµ(t, x, y): the fundamental solution of ∂u ∂t = ∆µu 71

Theorem 12. µ: voulme doubling with repsect to the resistance metric R, i.e. ∃C > 0, ∀x ∈ K, ∀r > 0, µ(BR(x, 2r)) ≤ Cµ(BR(x, r)) ⇒ ∃ a metric d, quasisymmetric to R, ∃β ≥ 2, ∀0 < t ≤ 1, ∀x, y ∈ K, pµ(t, x, y) ≤ c1 µ(Bd(x, t1/β)) exp

• − c2

d(x, y)β t 1/(β−1) , and if d(x, y)β ≤ c4t, then c3 µ(Bd(x, t1/β)) ≤ pµ(t, x, y) the structure (E, F) and µ ⇒ d giving a geometry of K 72