martin barlow department of mathematics university of british columbia · department of mathematics...

Analysis on the Sierpinski Carpet

Martin Barlow

Department of Mathematics

University of British Columbia

Analysis on the Sierpinski Carpet – p. 1/33

Summary.

Lecture 1.Background on ‘diffusions on fractals’. Basic properties ofthe Sierpinski carpet.

Lecture 2.Heat kernels on metric measure spaces.

Summary. The ‘analysis on fractals’ and ‘analysis on metric space’communities have tended to work independently.Metric spaces such as the Sierpinski carpet fail to satisfy many of theproperties which are generally assumed for metric spaces.


Percolation. (Broadbent and Hammersley (1957).) Take any graphG = (V,E) and a probabilityp ∈ (0, 1). For each edgee keepe withprobabilityp and delete it with probability1 − p, independently of allother edges.There are numerous applications in mathematical physical contexts,and this is one of the core models in statistical physics. Themodel isversatile, and can be applied in many other contexts – such ascontactnetworks for infectious diseases.Write Gp = (V,Ep) for the (random) graph obtained by thepercolation process. The connected component ofx is called theclustercontainingx and is denotedC(x).


p = 0.2


p = 0.2, largest cluster marked


p = 0.2


p = 0.4


p = 0.5(= pc)


p = 0.5


p = 0.6


p = 0.8


Percolation onZd has aphase transition:

Whenp is smallGp consists of lots of small clusters.Whenp is large there is one large (infinite) cluster.

Set

θ(p) = Pp(|C(0)| = ∞),

pc = pc(d) = inf{p : θ(p) > 0}.

Theorem 1 (Broadbent and Hammersley (1957)). For the latticeZd,

pc ∈ (0, 1).

Open Problem. Is θ(p) = 0 if 3 ≤ d ≤ 18? (True ford = 2, d ≥ 19.)


Physicists are interested in ‘transport’ problems of percolation clusters,i.e. how do solutions of equations such as the wave or heat equationbehave? A percolation cluster is a graph, so one can define thegraph(discrete) Laplacian

∆Gpf(x) =

1

Np(x)

∑

y∼px

(f(y) − f(x)).

Herey ∼p x means that{x, y} is an edge inGp, andNp(x) is thenumber of neighbours ofx.One can then look at the heat and wave equations onGp (discretespace, continuous time):

∂u

∂t= ∆Gp

u,∂2u

∂2t= ∆Gp

u.


Three phases:p < pc (subcritical). No large scale structure.p > pc (supercritical). For HE get Gaussian limits, homogenization.p = pc (critical). Hard and interesting, only a few results.

Of particular interest are critical exponents close topc. For example itis conjectured that

Ep|C(0)| ≈ (pc − p)−γ asp ↑ pc,

θ(p) ≈ (p − pc)β asp ↓ pc.

Universality. Physicists believe these exponents are ‘universal’. So ifthey can be calculated for percolation then the same values will holdfor more realistic models.More precisely: these exponents should depend on the dimension d,but not on the particular lattice. At first sight this seems a surprise,since coarser features of the model, such as the value ofpc do dependon the lattice.


P. De Gennes in a surveyLa percolation: un concept unificateur, LaRecherche 1976 , suggested the study of random walks on criticalpercolation clusters as a tool to study e.g. the heat equation on criticalclusters.It was believed then (and has now been proved in some cases) thatcritical percolation clusters are ‘fractal’.

To help understand how random walks would behave on such ‘fractal’graphs, in the early 1980s mathematical physicists studiedrandomwalks on regular exact fractals, such as the Sierpinski gasket.


The graphical Sierpinski gasket


Early 1980s: mathematical physicists studied ‘diffusion on fractals’,but really they looked at random walks on fractal graphs.

Late 1980s: mathematicians studied analysis on true fractals, startingwith the easiest case, the Sierpinski gasket. (S. Goldstein, S. Kusuoka,MB-E. Perkins, J. Kigami.)

1990–2005: mathematical work diverged from physics.Mathematicians obtained quite detailed information aboutsolutions ofthe heat equation on regular exact fractals such as the Sierpinski gasketand Sierpinski carpet. This work gave quite accurate results, butneeded very strong regularity for the space.Of more interest for physics applications (such as percolation) wouldhave been cruder results obtained under weaker hypotheses.

2005 on: mathematical theory developed tools which can handle someproblems for random walks on critical percolation clusters: MB, Jarai,Kumagai, Slade; Kozma, Nachmias.


I will discuss the work on the true fractals, which of course are alsometric spaces.

Sierpinski gasket: rather special because of existence of ‘cut points’.

Sierpinski carpet:harder, but forced us (MB and Richard Bass) todevelop more robust tools.

Sierpinski gasket and Sierpinski carpet


The basic Sierpinski carpet (SC) is a fractal subset of[0, 1]2 defined ina similar way to the classical Cantor set, except that one removes themiddle square out of a3 × 3 block.

The SC is a ‘model space’ for studying diffusion in irregularmedia.


In a similar fashion, one can define the basic SC ind dimensions, bydividing each cube into3d subcubes and removing the middle cube.One can also defineGeneralized Sierpinski carpetsin d ≥ 2 byremoving other (symmetric) patterns of cubes. For example theMenger sponge:

For simplicity, in these talks I will mainly discuss the basic SC inddimensions.


Call the SC (ind dimensions)F , and letFn be the thenthapproximation toF . Let LF = 3 (‘length scaling factor’) andMF = 3d − 1 (‘mass scaling factor’). ThenFn is the union ofMn

F

cubes each of sideL−nF , and it follows that the Hausdorff dimension of

F is

α = df (F ) = dimH(F ) =log MF

log LF.

Let µ be Hausdorffxα-measure onF , normalised so thatµ(F ) = 1.Lemma 2 Letx, y ∈ F . Then there exists a pathγ connectingx, yconsisting of countably many line segments, with length

L(γ) ≤ C|x − y|.

Thus ifd(x, y) is the shortest path (geodesic) distance onF then

|x − y| ≤ d(x, y) ≤ C|x − y|.


(F, d, µ) is a metric measure space, and one has

µ(B(x, r)) ≍ rα, x ∈ F, 0 < r < 1.

Upper gradients on the SC.

Definition. g : F → R is anupper gradient forf if for any x, y ∈ F andpath(γ(t), t ∈ [0, 1]) connectingx andy then

|f(y) − f(x)| ≤

∫

γg =

∫ 1

0g(γ(t))|γ′(t)|dt.

We write|∇f | for an upper gradient off .

In many situations a (weak)(q, p)–Poincaré inequality of the type

infa

∫

B(x,r)|f(x) − a|qµ(dx) ≤ C(r)

∫

B(x,λr)|∇f |pµ(dx)

gives useful information about the space.


Theorem 3 LetHi, i = 0, 1 be the left and right edges of the SC ind = 2: Hi = {x = (x1, x2) ∈ F : x1 = i}. Let

A = {f ∈ C(F ) : f |Hi= i, i = 0, 1}.

Letp > 0. Then there exists a sequence of functionsfn ∈ A such that

∫

F|∇fn|

pµ(dx) → 0.

Proof. Recall thatF0 = [0, 1]2 and thatFn is thenth step in theapproximation of the SC. Let

f0(x1, x2) = x1; so∫

F|∇f0|

pdµ = 1.


Chooseh1 : [0, 1] → [0, 1] to be continuous and linear on each interval[0, 1

3 ], [13 , 23 ], [23 , 1], with h(0) = 0, h(1) = 1, andh′

1 = b on

[0, 13 ] ∪ [23 , 1], andh′

1 = 3 − 2b on [13 , 23 ]. Set

f1(x1, x2) = h1(x1); note thatf ∈ A.

Then|∇f1| is equal tob on 6 out of the 8 subsquares, and(3 − 2b) on2 subsquares. Hence

∫

F|∇f1|

pµ(dx) = 68bp + 2

8(3 − 2b)p = 14(3bp + (3 − 2b)p) = ϕ(b).

Thenϕ(1) = 1 and (easy to check)ϕ′(1) = p/4 > 0. So there existsb ∈ (0, 1) with ϕ(b) < 1.With this choice ofb we have

∫

F|∇f1|

pdµ = ϕ(b) < 1.


We now iterate this construction, to obtain a sequencefk such that

∫

S|∇fk|

pdµ = ϕ(b)k → 0 ask → ∞. (1)

We definefk(x1, x2) = hk(x1) wherehk are defined by

hk+1(t) =

(b/3)hk(3t), 0 ≤ t ≤ 13

(b/3) + (1 − 2b/3)hk(3t − 1), 13 ≤ t ≤ 2

3

(1 − b/3) + (b/3)hk(3t − 2), 13 ≤ t ≤ 2

3 .

On each squareS side3−k we then have

∫

S|∇fk+1|

pdµ = ϕ(b)

∫

S|∇fk|

pdµ.

Hence ∫

F|∇fk|

pdµ = ϕ(b)k → 0 ask → ∞.


Suppose we define a quadratic formQ by

Q(f, f) =

∫

F|∇f |2dµ.

Corollary 4 Q is not closeable; there existsfn → 0 uniformly suchthatQ(fn − fm, fn − fm) → 0 butQ(fn, fn) ≥ 1 for all n.

So the pre-Hilbert spaceC(F ) with inner productQ(f, f) + ||f ||22 isnot closeable inL2(F ).We have that(fn) is Cauchy inH, andfn → 0 in L2. However,fn donot converge to 0 inH.


Proof. Let f be continuous onF . Using Theorem 3 we can find foreachn a functionhn such that

||f − hn||∞ ≤ 12 ||f ||∞, Q(hn, hn) ≤ 2−n.

(So continuous functions can be closely approximated by functions oflow energy.)

Let f0 satisfyQ(f0, f0) = 2, ||f0||∞ = C < ∞. Choose(gn) such thatQ(gn, gn) ≤ δ24−n and iffn+1 = fn − gn then||fn+1||∞ ≤ 1

2 ||fn||∞.Thenfn → 0 and

Q(fn+1, fn+1) = Q(fn, gn) − 2Q(fn, gn) + Q(gn, gn)

≥ Q(fn, fn) − 2δ2−nQ(fn, fn)1/2.

Choosingδ small enough we have1 ≤ Q(fn, fn) ≤ 3 for all n. �


So... what can one do? Letµn be Lebesgue measure onFn

renormalized to have mass 1. So (recallL = 3, M = 3d−1),

µn(dx) = 1Fn(x)(Ld/M)ndx.

The problem is that the quadratic/Dirichlet forms

Qn(f, f) =

∫

Fn

|∇f |2dµn

are ‘too small’. So we seek constantsan ↑ ∞ such that if

En(f, f) = anQn(f, f),

thenEn has (at least) subsequential limits.The obvious choice (which works) is to choosean so that

inf{En(f, f) : f ∈ A} = O(1).


Note thatFn consists ofMn cubes each of sideL−n. If we define

Fn = LnFn = {Lnx, x ∈ Fn},

thenFn is a subset of[0, Ln]d ⊂ Rd+ which is the union ofMn unit

cubes. We haveFn ∩ [0, Ln−1]d = Fn−1.

So setF = ∪∞

n=0Fn.

This is called thepre-Sierpinski carpet. It is the the closure of an opendomain inR

d with piecewise smooth boundary, so is locally regular. Ithas fractal structure ‘at infinity’, which mimics the local structure ofthe compact Sierpinski carpetF .Define also:

F = ∩∞

n=0L−nF .

This is theunbounded fractal Sierpinski carpet; it is the union of acountable number of copies ofF .


The renormalization argument involves various constants,and these

turn out to have a more intuitive form if we work with the ‘big’setsFn

rather than the ‘small’ setsFn.

Let Hn0 , Hn

1 be the left and right sides ofFn:

Hni = {x = (x1, . . . xd) ∈ Fn : x1 = Lni}, i = 0, 1,

andAn = {f ∈ C(Fn) : f |Hn

i= i, i = 0, 1}.

be the set of functions which are zero on the LHS and 1 on the RHS.Let

R−1n = inf{

∫

eFn

|∇f |2dx : f ∈ An}.

This is the minimal energy of functions inAn; and soRn can be

interpreted as the ‘effective resistance’ inFn betweenHn0 andHn

1 .


So it is natural to try

En(f, f) = L(d−2)nRn

∫

Fn

|∇f |2dx.

(The termLn(d−2) arises from rescaling the integral of the gradient.)

What one would like:En converge (in Mosco sense) to a limitingDirichlet formE .

All that can be proved so far: a compactness result which proves thatEn has subsequential limits.

Two main inputs:

1. Control of the constantsRn involved in the renormalization.

2. A Harnack inequality which gives enough regularity ofEn to givegood properties of the limit.


Theorem 5 There existsC such that

C−1RnRm ≤ Rn+m ≤ CRnRm, n,m ≥ 0.

Hence there existsρ such that

C−1ρn ≤ Rn ≤ Cρn, n ≥ 0.

Proof: R−1n is given as minimal energy of a function inAn. Use

copying and pasting of the optimal energy functions forR−1n andR−1

m

to construct a function inAn+m with energy less thancR−1n R−1

m .A dual characterization of resistance as the minimal energyof a unitflow gives an upper bound onRn+m.

The existence ofρ follows from general results on subadditivesequences; these give no information on the value ofρ.For the basic SC withd = 2 shorting/cutting arguments give7/6 ≤ ρ ≤ 3/2, and numerical calculations suggestρ ≃ 1.251.


The next input is an elliptic Harnack inequality (EHI) for the pre-SC.Note that the pre-SC is the closure of a domain inR

d, and so one can

define the Laplacian and harmonic functions onF in the usual way.

Write B(x, r) = {y : d(x, y) < r} for balls in the pre-SC with respectto the geodesic metric. (If we used the Euclidean metric we wouldhave to be careful about connectivity.)

Theorem 6 Letx ∈ F , r > 0, B = B(x, r) andh be non-negative onB and harmonic onB. There exists a constantCH (depending only onthe Sierpinski carpet) such that ifB′ = B(x, r/2) then

supB′

h ≤ CH infB′

h.

The proof uses a rather lengthy probabilistic coupling argument.


These two results lead to good control of the heat equation inF .

We have three ‘scale factors’ for the SC:1. L = LF , the length scaling factor.2. M = MF ,3. ρ, the ‘resistance scaling factor’.For the basic SC ind dimensions,L = 3, M = 3d − 1.For [0, 1]d (which can be regarded as a trivial SC) ifL = 3 thenM = 3d andρ = L2−d.Define:

df (F ) = α =log M

log L, dw(F ) = β =

log Mρ

log L.

df is the Hausdorff dimension, and gives the volume growth ratein F :

V (x, r) = |B(x, r)| ≍ rdf , qr ≥ 1.

dw was called by physicists thewalk dimension is is related to spacetime scaling of the heat equation.


Theorem 7 (MB+R. Bass) Letpt(x, y) be the heat kernel on the

pre-SCF . Then

pt(x, y)(c)≍ cV (x, t1/β)−1 exp(−c(d(x, y)β/t)1/(β−1)), (1)

for (t, x, y) such thatt ≥ 1 ∨ d(x, y).

1.(c)≍ means that upper and lower bounds hold, but with different

values of the constant on each appearance.2. One gets the usual Gaussian type bounds ift ≤ 1 ∨ d(x, y).

3. SinceF looks locally likeRd at length scales less than 1, one

expects different bounds whent, d(x, y) are small.4. Integrating these bounds, ifWt is the associated diffusion processone gets

Exd(x,Wt)2 ≍ t2/β , t ≥ 1.

Sinceβ > 2 this is called ‘anomalous diffusion’.


Introduce the conditions:

HK(β) – the heat kernel bounds in the Theorem.

RES(β): for anyx ∈ F , r ≥ 1 one has

Reff(B(x, r), B(x, 2r)c) ≍rβ

V (x, r).

(HereReff is the effective resistance between the two sets.)Theorem 8 ( Grigoryan and Telcs). The following are equivalent for ameasure metric space with Dirichlet form (MMD spaceX ) (1)Xsatisfies D=volume doubling, EHI andRES(β),(2) The heat kernel onX satisfiesHK(β).(More precise statement next lecture.)

We have seen that the pre-SC satisfies the conditions in (2).


Why does the parameterβ = log(Mρ)/ log L arise?

One way is to look at the Poincaré inequality in a cube inF .

Let S be a cube sideR = Ln. Let PS be the best constant in the PI∫

S(f − f)2dx ≤ PS

∫

S|∇f |2dx.

Let g be the function inA(S) which attains the minimum for theresistance acrossS. Then

∫

S(g − g)2 ≃ |S| = Mn,

∫

S|∇g|2 = R−1

n .

SoPS ≥ cMnRn ≃ c(Mρ)n = cRlog(Mρ)/ log L = cRβ .

The regularity given by EHI enable one to show that in factPS ≍ (Mρ)n.


Return to the fractal Sierpinski carpet.

The information about the pre-SC then enable one to prove that therescaled Dirichlet formsEn onFn have subsequential limits.Taking these limits, one obtains a Dirichlet form(E ,F) onL2(F, µ).Different subsequences might give different limits. However, any limitsatisfies symmetry conditions and has a heat kernel which satisfiesHK(β).A recent result (MB, Bass, Kumagai, Teplyaev) proves that any DFwhich is symmetric with respect to suitable local symmetries of the SCis unique, up to constants.


martin barlow department of mathematics university of british columbia · department of mathematics...

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