Christoph Bandt- Simple self-similar sets and Fractal analysis on infinitely ramified fractals

8/3/2019 Christoph Bandt- Simple self-similar sets and Fractal analysis on infinitely ramified fractals

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Simple self-similar sets and

Fractal analysis on

infinitely ramified fractals

Monastir, 12 September 2007

Christoph Bandt

Mathematics, University of Greifswald

[email protected]


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1.1 Fractal analysis

Classical analysis on fractal spaces.

Started with Alexander+Orbach and

Rammal+Toulouse, J. Physique 1982/83

Many names: Kusuoka, Lindstrom, Barlow,Fukushima, Kigami, Sabot, Kumagai ...

Various approaches:

Laplace operator, Dirichlet forms

Brownian motion Besov spaces harmonic functions resistance metric


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1.2 Main problem of fractal analysis

Well-developed theory, diverse methods.

But: very few examples,all looking like Sierpinski gasket.

A single point will disconnect the space,

at least locally.

Exception: existence of a Brownian motion

on Sierpinski carpets, proved by Barlow+Bass1990-99 (uniqueness, parameters, self-similarityremain open)With this lecture, we start analysis on a newclass of spaces.


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1.3 Why self-similar sets?

Take harmonic functions: they can be defined

in any metric space by the mean-value prop-

erty:

f(x) =1

(S)

Sf(y) d (y)

where S is a small sphere around x and is the

surface measure.

This makes sense if

there are many balls or ball-like sets with

similar structure (symmetry condition)

there is a concept of surface measure.

Both conditions hold on appropriate self-similar

sets: they form a good testbed for analysis.


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Overview

1. Motivation

2. Fractals of finite type

3. Fractal n-gons

4. Analysis on finitely ramified fractals

5. Resistance scaling on new spaces

6. Harmonic functions on the octagasket


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2.1 Self-similar sets

Mathematically tractable class of fractals

f1,...,fn contracting similitudes on Rn.

On C, fk

(z

) =kz

+ck

withrk

=|k| 1

then OSC does not hold.

Def. Mn = {| n-gon A() is connected} iscalled the Mandelbrot set for fractal n-gons.

Subset of (a quarter of) the unit disk.

M2 is well-known (Barnsley, Bousch,...).


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3.4 n-gon Mandelbrot sets

Mn has symmetry group D2n for odd nand symmetry group Dn for even n

r > 1n

Mn r sinn

1+sin n

M3with symmetry lines


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3.5 n-gon Mandelbrot sets

M4 with symmetry line

Geometric properties similar to M2.But no antenna on real axis.


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3.6 Hole in

M4

Computer construction of Mn with neighbormaps.


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3.7 OSC is uncountable

Few fractals with OSC are known

(Koch curve etc.). Only countably many?

Let D() = A1() A2()Th. (Solomyak 2005) There are uncountably

many for which the overlap set D() i s a

singleton.

Th. (Hung 2006) For every m N there areuncountably many for which OSC holds and

the overlap set D() consists of 2m points.

Th. (Hung + B 2006) For every [0, 0.2]there are uncountably many for which OSC

holds and the overlap set D() is a Cantor set

of Hausdorff dimension .

In each case, the addresses of the points ofD() are different for different .


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3.8 Classify Type 1 n-gons

For every n there are n3 parameters forwhich A() is type 1. The parameters can be

explicitly given. For odd n

1

= 1 +bk/2

2sin n

, 0 j <

n

6 +1

2 .

For even n, the formula is similar, but the in-

tersection sets may differ:

neighbors have one intersection point forall solutions with odd n

but only for half of solutions with even n the others have Cantor set intersections very few exceptions where pieces intersect

in other sets


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3.9 Cantor intersection set


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3.10 Other Cantor overlaps


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3.11 Example with piece overlap

This example has type 2 while

the golden gasket has type 4.


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3.12 Example with piece overlap

More complicated, but still finite type.


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4.1 P.c.f. Type 1 n-gons

Now we consider the type 1 n-gons with one-

point intersection between pieces. For odd n

there are n3 parameters .Here are all examples for n = 5.


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4.2 P.c.f. Type 1 n-gons

The addresses of the intersection point A0A1have the form 0k 1j.Let m = j 1 k and q = n (j 1).It turns out that the harmonic or resistance

structure of A depends only on k,m, and q.

Even n. 1 = 2 + i and 2 +

2 + i


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4.3 Resistances for type 1

The 3 boundary points are always in the po-

sition X = 0, Y = k, and Z = j 1, up tosymmetry.

This allows to scale resistances:

RX = rx, RY = ry, RZ = rz

X

Y Z

R

RRY Z

Xx

z

r

X

Y Z

y

YZ

X

k

m

qx

x

r

zy

r

xx

xrxxxxrxxx

xrxxx xrxxx

xrxxx

xrxxx

zr

zr

yr yr

a b c d


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4.4 Result for p.c.f. type 1

Th. (Hung + B 2007) For all p.c.f. type 1

fractal n-gons,

=1

2n

n + M +

(M n)2 + 4n(K+ Q M)

where K = k(n k), M = m(n m) andQ = q(n q).

Different resistance exponents = log log ||.

We obtained

[0.76, 0.82] with fractal di-

mensions in [1.63, 1.76] for the examples shown

above. For the Sierpinski gasket, = 0.74 and

dH = 1.58.


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5.1 The octagasket

Fractal 8-gon with = 1

22 . Type 1!

Intersections are linear Cantor sets generated

by two maps with factor . So their dimension

is 14 dim A.


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5.2 The octagaskets resistance

Three sets X , Y , Z where A can touch a neigh-

bor. X Y Z is the boundary of A.

X

Y Z

There are only two possible types of pairs of

neighbors: XY and XZ.

Thus only two equations for the normalization

of resistances have to be solved.


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5.3 The resistance scheme

Network with nodes x,y,z for a pieces Ak and

X , Y , Z for the set A.

Normalize effective resistance:

xz = yz = 1, xy = 2r

r

r

1-r

For the large network, determine XY, XZ byKirchhoffs laws and solve

XY = xy, XZ = xz for and r.


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5.4 Resistance exponent

The resistance factor for the octagasket is

=1 +

193

8 1.86

That is, for a set which is 1

|| 3.14 larger,

the resistance increases by 1.86.

The resistance exponent is

=log

log || 0.506

and the dimension dH =

log8

log || 1.69 .Compare with pentagasket:

=9 +

161

10 2.17, 0.80, dH 1.67


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5.5 Resistance metric

Th. On the octagasket there exist a self-

similar effective resistance metric which gen-

erates the topology of A.

This metric is unique if distances of boundarysets (not only points) are taken into account.


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5.6 Two other examples

Resistance scaling can be done with other

infinitely ramified sets.

However, calculations become more involved.

For the fractal with four pieces, a fifth degree

polynomial comes in, and

1.56,

0.54, dH

1.67 .

For the dodecagasket the numerical result is

2.93, 0.69, dH 1.60 .


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6.1 Dirichlet on octagasket

Dirichlet problem on the octagasket:

given boundary values 0(x) on X,Y,Z, Holder

continuous with respect to the Cantor struc-

ture, find a harmonic function (x) on A ex-

tending 0.

What is a harmonic function?

Mean-value properties.

First construct one value aw for each piece Aw

of A such that

aw =1

8

8i=1

awi

If this goes through, aw is the mean value of

with respect to the Hausdorff measure

H, the

natural volume on A.


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6.2 The mean value

Writing a for the mean on A, we require

a =1

8

8i=1

ai

We want to express a as a mean of the bound-ary values bx, by, bz which are assumed to be

constant. By symmetry,

a = q(bx + by) + ( 1 2q)bzIt turns out that

q =331 193

1008 0.31459

is not related to the resistance

r =17 193

8 0.38844


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6.3 Harmonic functions

Def. A bounded real function on A is Holder

continuous if for some < 1

diam (Aw) m for all w = w1..wm

Such a function is uniquely determined by itsmean values

aw =1

H(Aw)

Aw d H

for all words w = w1..wm.

The aw are kind of wavelet coefficients of .Def. A Holder function is harmonic if

aw = q(bxw + b

yw) + ( 1 2q)bzw

for all w where bxw, byw, b

zw denote the average

of on the three boundary sets of Aw, andq = 0.314... is the constant above.


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6.5 Refining solutions

Suppose we have for a 0 which is constant

on X , Y , Z say 0 = 0 for simplicity and

now we want to introduce two different values

bx0 = 0(X0) = and bx1 = 0(X1) = 1 on

the two parts of X. Then the coefficient a of remains unchanged iff

=428 193

337 1.23

This can be reformulated as follows:

if we want to refine 0 by adding symmetriccorrections +, on X0, X1 then

bx0 bx = (1 q)(+) , bx1 bx = q()where

q =

1 + 0.551


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6.6 Harmonic measure

The constant q expresses the greater impor-tance of X1 : For random walkers starting on A

(with respect to H) and hitting the boundaryat X, the point will be in X1 with probability

q, and in X0 with 1 q.This interpretation (which should be a point

for defining Brownian motion on A) gives rise

to the definition of a harmonic measure.

Th. The harmonic measure on B can be ex-plicitly described: it has an alternating product

structure with q, 1 q on the sets X,Y,Z.The bxw, b

yw, b

zw are taken as averages with re-

spect to this measure.

The proofs are still in work.

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Christoph Bandt- Simple self-similar sets and Fractal analysis on infinitely ramified fractals