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Fractal Geometry,Graph and Tree

Constructions

Tommy Lofstedt

Outline

ProblemDescription

Fractal Geometry

Graph-DirectedConstructions

Tree Constructions

Equivalence of Graph and TreeConstructions

Equivalence forUnion

Results andConclusions

Fractal Geometry, Graph and TreeConstructions

Master’s Thesis Presentation

Tommy Lofstedt

Department of Mathematics and Mathematical StatisticsUniversity of Umea

[email protected]

February 8, 2008

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Constructions

Tommy Lofstedt

Outline

ProblemDescription

Fractal Geometry


Tree Constructions




Outline of presentation

Problem Description

Fractal Geometry

Graph-Directed Constructions

Tree Constructions

Equivalence of Graph and Tree Constructions

Equivalence for Union

Results and Conclusions

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Constructions

Tommy Lofstedt

Outline

ProblemDescription

Fractal Geometry


Tree Constructions




Problem description

Previous research

Goals

Purpose Methods

Graph-directed Constructions

Tree Constructions

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Constructions

Tommy Lofstedt

Outline

ProblemDescription

Fractal Geometry


Tree Constructions




Fractal geometry

∼2400 km

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Constructions

Tommy Lofstedt

Outline

ProblemDescription

Fractal Geometry


Tree Constructions




Fractal geometry

∼2800 km

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Constructions

Tommy Lofstedt

Outline

ProblemDescription

Fractal Geometry


Tree Constructions




Fractal geometry

∼3450 km

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Constructions

Tommy Lofstedt

Outline

ProblemDescription

Fractal Geometry


Tree Constructions




Fractal geometry – What is a fractal?

Description:

A fractal is a rough or fragmented geometric shape that can be subdivided in parts, each of which is (at least approximately) a reduced/size copy of the whole.

– Benoıt Mandelbrot, 1975

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Constructions

Tommy Lofstedt

Outline

ProblemDescription

Fractal Geometry


Tree Constructions





Definition The set has fine structure, it has details on arbitrary

scales.

The set is too irregular to be described with classical

euclidean geometry, both locally and globally. The set has some form of self-similarity, this could be

approximate or statistical self-similarity.

The Hausdorff dimension of the set is strictly greater

than its Topological dimension. The set has a very simple definition, i.e. it can be

defined recursively.

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Constructions

Tommy Lofstedt

Outline

ProblemDescription

Fractal Geometry


Tree Constructions





Example (The Cantor set)

E 0:

E 1: E 2:

. . .

E ∞:

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Constructions

Tommy Lofstedt

Outline

ProblemDescription

Fractal Geometry


Tree Constructions





Example (The von Koch curve)

E 0:

E 1:

E 2:

. . .

E ∞:

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Constructions

Tommy Lofstedt

Outline

ProblemDescription

Fractal Geometry


Tree Constructions




Fractal geometry – Similarity dimension

Magnify a unit line segment twice

Magnify a unit square twice

Magnify a unit cube twice

In general we have mD = N . Solving for D gives the

dimension as:

D =log N

log m.

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Constructions

Tommy Lofstedt

Outline

ProblemDescription

Fractal Geometry


Tree Constructions




Fractal geometry – Hausdorff dimension

The Hausdorff dimension: The most general notion

Defined for all sets in metric spaces

Requires some measure theory

The notion of a measure: Ascribe a numerical size to a set

A function µ : S → [0, ∞]

µ(∅) = 0

If A ⊆ B then µ(A) ≤ µ(B )

µ (∞

i =1 Ai ) ≤∞

i =1 µ(Ai )

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Constructions

Tommy Lofstedt

Outline

ProblemDescription

Fractal Geometry


Tree Constructions





The Hausdorff measure has all these properties, and isdefined as:

Definition

Let F be a subset of a metric space, e.g. Rn, and s ∈ R+,then for any δ > 0 we let:

H s δ (F ) = inf

∞i =1

diam(U i )s : {U i } is a δ−cover of F

When δ decreases, H s δ (F ) increases, and approaches a limit

as δ → 0. We write:

H s (F ) = limδ→0

H s δ (F ).

We call H s (F ) the s-dimensional Hausdorff measure of F.

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Constructions

Tommy Lofstedt

Outline

ProblemDescription

Fractal Geometry


Tree Constructions





If H s (F ) < ∞ then H t (F ) = 0 when t > s . There is acritical value of s at which H s (F ) jumps from ∞ to 0. This

value is known as the Hausdorff dimension of F , and isdenoted dimH F .

F l GF l B d

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Constructions

Tommy Lofstedt

Outline

ProblemDescription

Fractal Geometry


Tree Constructions




Fractal geometry – Box-counting dimension

The Box-counting dimension:

The most common definition in practical use

Easy to calculate mathematically

Easy to estimate empirically

F t l G tF l B i di i

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Constructions

Tommy Lofstedt

Outline

ProblemDescription

Fractal Geometry


Tree Constructions





Note that:

The number of line segments of length δ that areneeded to cover a line of length l is l /δ

The number of squares with side length δ that areneeded to cover a square with area A is A/δ2

The number of cubes with side length δ that are neededto cover a cube with volume V are V /δ3

The dimension of the object we try to cover is the

power of the side length, δ, of the box used

Fractal GeometryF l B i di i

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Constructions

Tommy Lofstedt

Outline

ProblemDescription

Fractal Geometry


Tree Constructions





Let the number of boxes needed to cover a set F be N δ(F ).

Following the discussion above, the number of boxes neededto cover the object should be proportional to the box size:

N δ(F ) ∼C

δs , δ → 0

limδ→0

N δ(F )

δ−s = C .

limδ→0

(log N δ(F ) + s log δ) = log C .

s = limδ→0

log N δ(F ) − log C

− log δ= lim

δ→0

log N δ(F )

− log δ.

Fractal GeometryF t l t B ti di i

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Constructions

Tommy Lofstedt

Outline

ProblemDescription

Fractal Geometry


Tree Constructions





More formally:

Definition

The lower and upper Box-counting dimensions of a set F aredefined as

dimBF = lim inf δ→0

log N δ(F )

− log δ

dimBF = lim supδ→0

log N δ(F )

− log δ,

respectively. If their values are equal, we refer to the

common value as the Box-counting dimension of F

dimB F = limδ→0

log N δ(F )

− log δ.

Fractal GeometryG h Di t d C t ti

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Constructions

Tommy Lofstedt

Outline

ProblemDescription

Fractal Geometry


Tree Constructions





The Cantor set, C , is constructed using two similaritytransformations, S 1(x ) = x

3 and S 2(x ) = x 3 + 2

3 . Then

C = S 1(C ) ∪ S 2(C ) and

H s (C ) = H s (S 1(C )) + H s (S 2(C )).

By the scaling property of the Hausdorff measure we have

H s (C ) =

1

3

s

H s (C ) +

1

3

s

H s (C ) = 2

1

3

s

H s (C ).

This means that

1 = 21

3s

and therefore

s =log2

log3,

the Hausdorff dimension of C , the Cantor set.

Fractal GeometryGraph Directed Constr ctions

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Constructions

Tommy Lofstedt

Outline

ProblemDescription

Fractal Geometry


Tree Constructions





U 0 V 0 U 1 V 1 U 2 V 2

U 3 V 3 U 4 V 4 U V

Consider a case where

U = S 1(U ) ∪ S 2(V )and

V = T 1(U ) ∪ T 2(V ).

The Hausdorff measure of U and V is then

H s (U ) = H s (S 1(U )) + H s (S 2(V ))and

H

s

(V ) = H

s

(T 1(U )) + H

s

(T 2(V )).

Fractal Geometry,Graph Directed Constructions

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G y,Graph and Tree

Constructions

Tommy Lofstedt

Outline

ProblemDescription

Fractal Geometry


Tree Constructions





By the scaling property of the Hausdorff measure we get

H s (U ) = r s 1 H s (U ) + r s 2 H s (V )

andH s (V ) = r s 3 H s (U ) + r s 4 H s (V ).

This is a linear relationship, and therefore we can write theabove as

v = H s (U )H s (V ) , M =

r s 1 r s 2

r s 3 r s 4 ,

which gives the neat matrix equation

v = Mv.


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yGraph and Tree

Constructions

Tommy Lofstedt

Outline

ProblemDescription

Fractal Geometry


Tree Constructions





Generalizes to directed multigraphs, G = (V , E ) Hausdorff and Box-counting dimensions can be found

directly

Each vertex, v ∈ V , corresponds to a metric space

Each edge corresponds to a similarity S e , withcontraction r e

The dimension of such graph-directed sets is given byan the n × n adjacency matrix with elements

A(s )i , j =

e ∈E i , j

r s e .


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Graph and TreeConstructions

Tommy Lofstedt

Outline

ProblemDescription

Fractal Geometry


Tree Constructions





A set that is described by the graph

has adjacency matrix

A(s ) = 12s 1

4s

12s

34s .


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Tommy Lofstedt

Outline

ProblemDescription

Fractal Geometry


Tree Constructions





The spectral radius of A(s

), is the largest positive eigenvalueof the matrix:

ρ(A(s )) = max1≤i ≤n

λi .

As we saw before, the Hausdorff dimensions of the sets is the

eigenvector corresponding to eigenvalue 1.

It turns out that the value of s for which we have spectralradius 1 and a corresponding eigenvector with all positiveelements is the Hausdorff dimension of the underlying set,

i.e.dimH F = s .

Fractal Geometry,G h dGraph-Directed Constructions

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Tommy Lofstedt

Outline

ProblemDescription

Fractal Geometry


Tree Constructions





This is formalized in the following theorems:

TheoremLet E 1, . . . , E n be a family of graph-directed sets, and {T (i , j )}, be a strongly connected similarities without overlaps. Then there is a number s such that dimH E i = dimB E i = dimB E i = s and 0 < H s (E i ) < ∞ for

all i = 1, . . . , n. Also, s is the unique number satisfying ρ(A(s )) = 1.

Theorem

Each graph-directed construction has dimensions = max{s H : H ∈ SC (G )}, where s H is the unique number such that ρ(H (s H )) = 1. The construction object, F, has positive and σ-finite H s measure. This number s is suchthat dimH F = dimB F = dimB F = s.

Fractal Geometry,G h d TTree Constructions

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Tommy Lofstedt

Outline

ProblemDescription

Fractal Geometry


Tree Constructions




Tree Constructions

An alphabet of N letters is denoted asE = {0, . . . , N − 1}

A string is a series of letters from E , e.g. 10110

The length of the string and is written as |α|

There is a unique string of length 0, denoted by Λ

α n is the prefix of α of length n

Write α ≤ β if |α| ≤ |β | and β = αγ

Let [α] be the set of all strings with prefix α

The edges of a tree are labeled with the symbols of thealphabet

The strings are paths in the tree and Λ is the root

Let E (n) be the tree of all string of length n

Let E (∗) be the tree of all finite strings

Let E (ω) be the set of all infinite strings

Fractal Geometry,G h d T eeTree Constructions

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Tommy Lofstedt

Outline

ProblemDescription

Fractal Geometry


Tree Constructions




Tree Constructions

Let real numbers w α be given for each node α of the tree

E (∗), such that

w α > w β when α < β

and

lim|α|→∞w α = 0 for α ∈ E (∗)

.

A metric ρ is now defined as follows:

If σ = τ then ρ(σ, τ ) = 0

If σ = τ , then ρ(σ, τ ) = w α, where α is the longestcommon prefix of σ and τ

Then ρ is a metric on E (ω) such that diam[α] = w α for all α.

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Fractal Geometry,Graph and TreeTree Constructions




Tommy Lofstedt

Outline

ProblemDescription

Fractal Geometry


Tree Constructions




Tree Constructions

If we have several recurrent subsets, as in the graph-directed

case, we can let one tree represent each set, called pathspaces.

Metrics are defined for each path space, so thatdiam[α] = w α, as above. The numbers w α are defined as

w Λv = q v ,

w e α = r e w α,

with r e being the contraction ratio of a similaritycorresponding to an edge e between nodes in the tree andwhere q v is the diameter of the tree rooted at v .

Fractal Geometry,Graph and TreeTree Constructions

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Tommy Lofstedt

Outline

ProblemDescription

Fractal Geometry


Tree Constructions




If we select the q v such that

(diam[α])s =α→αe

(diam[αe ])s , (1)

for some s , then there exists a measure on each of thespaces satisfying µ([α]) = (diam[α])s for all α ∈ E

(∗)v .

Theorem

If ρ is a metric on E (ω)v and s > 0 satisfy

µ([α]) = (diam[α])s for all α ∈ E (∗)v , then µ(B ) = H s (B )

for all Borel sets B ⊆ E (ω)v .

Thus, H s (E (ω)v ) = µ(E

(ω)v ) = q s v and since 0 < q v < ∞ we

have dimH E (ω)v = s .

Fractal Geometry,Graph and TreeEquivalence of Graph and Tree Constructions

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Tommy Lofstedt

Outline

ProblemDescription

Fractal Geometry


Tree Constructions




q p

Rearranging Equation 1 we see that

q s u =e ∈E uv v a tree

r s e · q s v ,

for all trees u .

Expanding the above equation for each tree v 1, . . . , v n we get

q s v 1 = r s e v 1v 1· q s v 1 + · · · + r s e v 1v n

· q s v n...

q s v n = r s e v nv 1· q s v 1 + · · · + r s e v nv n · q s v n


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pConstructions

Tommy Lofstedt

Outline

ProblemDescription

Fractal Geometry


Tree Constructions




q p

Rewrite this in matrix form and we get

q s v 1q s v 1

...q s v

1

=

r s e v 1v 1r s e v 1v 2

. . . r s e v 1v nr s e v 2v 1

r s e v 2v 2. . . r s e v 2v n

......

. . ....

r s e v nv 1

r s e v nv 2

. . . r s e v nv n

q s v 1q s v 1

...q s v

1

.

Hence, by the above theorem, we can rewrite the equation as

H s (v 1)

H

s

(v 2)...H s (v n)

=


. . . r s e v 1v nr s e v 2v 1

r s e v 2v 2

. . . r s e v 2v n...

.... . .

...r s e v nv 1

r s e v nv 2. . . r s e v nv n

H s (v 1)

H

s

(v 2)...H s (v n)

.


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Constructions

Tommy Lofstedt

Outline

ProblemDescription

Fractal Geometry


Tree Constructions




q p

That equation can in turn be rewritten as

v = A(s )v ,

where

A(s ) =

r s

e v 1v 1 r s

e v 1v 2 . . . r s

e v 1v n


. . . r s e v 2v n...

.... . .

...r s e v nv 1

r s e v nv 2. . . r s e v nv n

, v =

H s

(E

(ω)

v 1 )H s (E

(ω)v 2 )

...

H s (E (ω)v n )

.

We note that v is an eigenvector of A(s ) with eigenvalue 1,and conclude that this is equivalent to the graph-directedsolution we saw before.

Fractal Geometry,Graph and TreeEquivalence for Union

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Constructions

Tommy Lofstedt

Outline

ProblemDescription

Fractal Geometry


Tree Constructions




The Hausdorff and upper Box-dimensions are finitely stable :

Theorem

dimk

i =1

F i = max1≤i ≤k

dim F i ,

for any finite collection of sets {F 1, . . . , F k }.

The Hausdorff dimension is also countably stable , i.e.:

Theorem

If F 1

, F 2

, . . . is a countable sequence of sets, then

dimH

∞i =1

F i = sup1≤i <∞

dimH F i .


C


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Constructions

Tommy Lofstedt

Outline

ProblemDescription

Fractal Geometry


Tree Constructions




This can be done equally well for with Graph-Directed

Constructions (and with Tree Constructions). Consider theunion of two graphs G 1 = (V 1, E 1) and G 2 = (V 2, E 2). Thisis

G = G 1 ∪ G 2 = (V 1 ∪ V 2, E 1 ∪ E 2) = (V , E ).


C i


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Constructions

Tommy Lofstedt

Outline

ProblemDescription

Fractal Geometry


Tree Constructions




We have the following theorem:

TheoremLet G i be graph-directed constructions. For the union graph

G =n

i =1

G i = (n

i =1

V i ,n

i =1

E i )

it holds that

s = max{s H : H ∈ SC (G )},

where s H is the unique number such that ρ(H (s H )) = 1. The construction object, F, has positive and σ-finite H s measure.The number s is such that dimH F = dimB F = dimB F = s.


C t ti


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Constructions

Tommy Lofstedt

Outline

ProblemDescription

Fractal Geometry


Tree Constructions




We can trivially deduce the following corollary:

Corollary

Let two graphs, G 1 and G 2 be represented as adjacency matrices, A1 and A2, respectively. The union graph,G 1 ∪ G 2, then has the adjacency matrix

A =

A1 00 A2

,

i.e., the block-diagonal matrix with A1 and A2 on the diagonal. Let

A(s ) =

A(s )

1 00 A

(s )2

.

Then the value of s for which ρ(A(s )) = 1 is such that dimH F = dimB F = dimB F = s.


Constructions

Results and Conclusions

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Constructions

Tommy Lofstedt

Outline

ProblemDescription

Fractal Geometry


Tree Constructions




Classical theory is just a special case of the

Graph-Directed Constructions Result for graph union is never seen elsewhere

Tree Constructions are equivalent to Graph-DirectedConstructions

Are there equally simple results for intersection,product, etc?

There is a very close relationship between fractalgeometry, graph theory and linear algebra. Can results

from these areas be used in fractal geometry? Using multifractal theory is the natural next step to

describe measures with Graph-Directed constructions


Constructions

The End

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Constructions

Tommy Lofstedt

Outline

ProblemDescription

Fractal Geometry


Tree Constructions




Thank you for listening!

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Pres Graph Tree (1)