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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
February 8, 2008
8/3/2019 Pres Graph Tree (1)
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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
Outline of presentation
Problem Description
Fractal Geometry
Graph-Directed Constructions
Tree Constructions
Equivalence of Graph and Tree Constructions
Equivalence for Union
Results and Conclusions
8/3/2019 Pres Graph Tree (1)
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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
Problem description
Previous research
Goals
Purpose Methods
Graph-directed Constructions
Tree Constructions
8/3/2019 Pres Graph Tree (1)
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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
∼2400 km
8/3/2019 Pres Graph Tree (1)
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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
∼2800 km
8/3/2019 Pres Graph Tree (1)
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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
∼3450 km
8/3/2019 Pres Graph Tree (1)
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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 – 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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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 – What is a fractal?
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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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 – What is a fractal?
Example (The Cantor set)
E 0:
E 1: E 2:
. . .
E ∞:
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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 – What is a fractal?
Example (The von Koch curve)
E 0:
E 1:
E 2:
. . .
E ∞:
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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 – 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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8/3/2019 Pres Graph Tree (1)
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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 – 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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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 – Hausdorff dimension
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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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 – Hausdorff dimension
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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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 – 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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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 – Box-counting dimension
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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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 – Box-counting dimension
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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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 – Box-counting dimension
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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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
Graph-Directed 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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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
Graph-Directed 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
Graph-DirectedConstructions
Tree Constructions
Equivalence of Graph and TreeConstructions
Equivalence forUnion
Results andConclusions
Graph-Directed 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.
Fractal Geometry,Graph Directed Constructions
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yGraph and Tree
Constructions
Tommy Lofstedt
Outline
ProblemDescription
Fractal Geometry
Graph-DirectedConstructions
Tree Constructions
Equivalence of Graph and TreeConstructions
Equivalence forUnion
Results andConclusions
Graph-Directed 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 .
Fractal Geometry,Graph Directed Constructions
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Graph and TreeConstructions
Tommy Lofstedt
Outline
ProblemDescription
Fractal Geometry
Graph-DirectedConstructions
Tree Constructions
Equivalence of Graph and TreeConstructions
Equivalence forUnion
Results andConclusions
Graph-Directed Constructions
A set that is described by the graph
has adjacency matrix
A(s ) = 12s 1
4s
12s
34s .
Fractal Geometry,Graph Directed Constructions
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Graph and TreeConstructions
Tommy Lofstedt
Outline
ProblemDescription
Fractal Geometry
Graph-DirectedConstructions
Tree Constructions
Equivalence of Graph and TreeConstructions
Equivalence forUnion
Results andConclusions
Graph-Directed 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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Graph and TreeConstructions
Tommy Lofstedt
Outline
ProblemDescription
Fractal Geometry
Graph-DirectedConstructions
Tree Constructions
Equivalence of Graph and TreeConstructions
Equivalence forUnion
Results andConclusions
Graph-Directed 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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Graph and TreeConstructions
Tommy Lofstedt
Outline
ProblemDescription
Fractal Geometry
Graph-DirectedConstructions
Tree Constructions
Equivalence of Graph and TreeConstructions
Equivalence forUnion
Results andConclusions
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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Graph and TreeConstructions
Tommy Lofstedt
Outline
ProblemDescription
Fractal Geometry
Graph-DirectedConstructions
Tree Constructions
Equivalence of Graph and TreeConstructions
Equivalence forUnion
Results andConclusions
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
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Graph and TreeConstructions
Tommy Lofstedt
Outline
ProblemDescription
Fractal Geometry
Graph-DirectedConstructions
Tree Constructions
Equivalence of Graph and TreeConstructions
Equivalence forUnion
Results andConclusions
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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Graph and TreeConstructions
Tommy Lofstedt
Outline
ProblemDescription
Fractal Geometry
Graph-DirectedConstructions
Tree Constructions
Equivalence of Graph and TreeConstructions
Equivalence forUnion
Results andConclusions
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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Graph and TreeConstructions
Tommy Lofstedt
Outline
ProblemDescription
Fractal Geometry
Graph-DirectedConstructions
Tree Constructions
Equivalence of Graph and TreeConstructions
Equivalence forUnion
Results andConclusions
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
Fractal Geometry,Graph and TreeEquivalence of Graph and Tree Constructions
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pConstructions
Tommy Lofstedt
Outline
ProblemDescription
Fractal Geometry
Graph-DirectedConstructions
Tree Constructions
Equivalence of Graph and TreeConstructions
Equivalence forUnion
Results andConclusions
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 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
H s (v 1)
H
s
(v 2)...H s (v n)
.
Fractal Geometry,Graph and TreeEquivalence of Graph and Tree Constructions
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Constructions
Tommy Lofstedt
Outline
ProblemDescription
Fractal Geometry
Graph-DirectedConstructions
Tree Constructions
Equivalence of Graph and TreeConstructions
Equivalence forUnion
Results andConclusions
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 1r 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
, 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
Graph-DirectedConstructions
Tree Constructions
Equivalence of Graph and TreeConstructions
Equivalence forUnion
Results andConclusions
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 .
Fractal Geometry,Graph and Tree
C
Equivalence for Union
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Constructions
Tommy Lofstedt
Outline
ProblemDescription
Fractal Geometry
Graph-DirectedConstructions
Tree Constructions
Equivalence of Graph and TreeConstructions
Equivalence forUnion
Results andConclusions
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 ).
Fractal Geometry,Graph and Tree
C i
Equivalence for Union
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Constructions
Tommy Lofstedt
Outline
ProblemDescription
Fractal Geometry
Graph-DirectedConstructions
Tree Constructions
Equivalence of Graph and TreeConstructions
Equivalence forUnion
Results andConclusions
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.
Fractal Geometry,Graph and Tree
C t ti
Equivalence for Union
8/3/2019 Pres Graph Tree (1)
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Constructions
Tommy Lofstedt
Outline
ProblemDescription
Fractal Geometry
Graph-DirectedConstructions
Tree Constructions
Equivalence of Graph and TreeConstructions
Equivalence forUnion
Results andConclusions
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.
Fractal Geometry,Graph and Tree
Constructions
Results and Conclusions
8/3/2019 Pres Graph Tree (1)
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Constructions
Tommy Lofstedt
Outline
ProblemDescription
Fractal Geometry
Graph-DirectedConstructions
Tree Constructions
Equivalence of Graph and TreeConstructions
Equivalence forUnion
Results andConclusions
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
Fractal Geometry,Graph and Tree
Constructions
The End
8/3/2019 Pres Graph Tree (1)
http://slidepdf.com/reader/full/pres-graph-tree-1 40/40
Constructions
Tommy Lofstedt
Outline
ProblemDescription
Fractal Geometry
Graph-DirectedConstructions
Tree Constructions
Equivalence of Graph and TreeConstructions
Equivalence forUnion
Results andConclusions
Thank you for listening!