DIMENSION

OF STRICTLY SELF SIMILAR SETS

A Thesis

Presented to the

Faculty of

California State Polytechnic University, Pomona

In Partial Fulfllment

Of the Requirements for the Degree

Master of Science

In

Mathematics

By

Santiago Echeverria

2018 SIGNATURE PAGE

THESIS: DIMENSION OF STRICTLY SELF SIMILAR SETS

AUTHOR: Santiago Echeverria

DATE SUBMITTED: Spring 2018

Department of Mathematics and Statistics

Dr. Robin Wilson Thesis Committee Chair Mathematics & Statistics

Dr. John A. Rock Mathematics & Statistics

Dr. Alan Krinik Mathematics & Statistics

ii ACKNOWLEDGMENTS

Its been a long way since I started as a mere community college student who only thought was nothing more than numbers. I can only thank my family and my friends who were always believing in me and helpingalong the way to this moment.

These people have made me into the person I am today and thank them for their support in this long and tiring journey. Especially, my professor Patricia Hale who been very patient with me and helped me with any questions or concerns I have had since I started this paper.

iii ABSTRACT

In this paper, many concepts and ideas will be presented about fractals and their dimension. There will be a overlook of a few defnitions which classify fractals and what type of dimension they possess. The types of fractals which will be covered are sets which we will classify as strictly and non-strictly self similar sets. Also there will be a comparison of two di˙erent dimensions on a strictly self similar set.

iv Contents

Signature Page ii

Table of Contents vi

List of Figures viii

1 Introduction 1

2 Basic Defnitions 4

2.1 Logarithms and Exponential Laws ...... 4

2.2 Topology ...... 5

2.3 Functions ...... 8

2.4 Fractal Defnitions ...... 12

2.5 Strictly Self Similar Sets ...... 15

2.5.1 The Cantor Set ...... 16

2.5.2 The Sierpinski Triangle ...... 16

2.5.3 The Von Koch Curve ...... 19

3 Non-Similar Fractals 23

3.1 The Julia Sets ...... 23

v 3.2 The Mandelbrot Set ...... 25

4 Dimension 30

4.1 Topological Dimension ...... 31

4.2 Similarity Dimension ...... 32

4.3 Hausdor˙ Dimension ...... 35

4.3.1 Hausdor˙ Measure ...... 35

5 Comparing Dimension 42

References 49

vi List of Figures

1.1 The Cantor Set [2] ...... 2

1.2 The Cantor Set ...... 3

2.1 Similar Triangles [5] ...... 12

2.2 Diagram of a Fern [4] ...... 14

2.3 A Shoreline [20] ...... 14

2.4 Waclaw Sierpinski [16] ...... 17

2.5 The Sierpinski Triangle [15] ...... 18

2.6 Open set under IFS ...... 19

2.7 The Von Koch Curve [7] ...... 21

2.8 Helge Von Koch [11] ...... 21

2.9 Open Trapezoid under IFS ...... 22

3.1 The Mandelbrot Set [17] ...... 25

3.2 Main Cardioid, Primary bulbs, Spokes, and Main antennas [6] . . . 27

3.3 4th primary bulb on the Mandelbrot Cactus [6] ...... 28

3.4 Smallest antenna on the 4th primary bulb [6] ...... 28

3.5 Fibonacci Numbers Found in the Mandelbrot set [6] ...... 29

4.1 Unit Cube [1] ...... 33

vii 4.2 The Sierpinski Triangle ...... 36

4.3 Closed balls covering boundary of unit square...... 38

4.4 The Hausdor˙ dimension is the value where jump discontinuity oc-

curs [3] ...... 39

n 5.1 The sets En = Vm where n = 0, 1, 2 ...... 45

i 5.2 C ∩ Vj where i = 0, 1, 2 ...... 46

viii Chapter 1

Introduction

The understanding of a set containing some similar pat-

tern or relation throughout its structure is the study of

fractals. Fractals appear in many shapes and forms in

our own world such as in plants, animals, and in math-

ematical objects. These fractals are becoming more and

more popular by mathematicians who wish to fnd their

secrets which are hidden in plain sight. These secrets

could be just how two fractals relate or how to product Georg Cantor [18] such an object. Before we start to give answers to these

topics we need to know what fractals are and how they are constructed.

When Georg Cantor (1845-1918) frst started to lay the foundations for modern

day set theory, he introduced one of the most important fractals in his paper pub-

lished in 1883 called the Cantor Set. The Cantor Set is a fractal and is an infnite set of points which are usually considered to lie in the closed interval [0,1] or unit interval. The mannerin which we constructthe Cantor Set is by an infniteprocess

1 that follows the following steps:

• Step 0: Consider the closed interval F0 = [0, 1].

• Step 1: Remove the middle 3rd of the interval [0, 1] and so we are left with

1 2 [0, 3 ]∪[ 3 , 1]. Note that at each of these two intervals are similarto the original 1 with similarity constant 3 and call this new set F1.

rd • Step 2: Remove the middle 3 from each interval in F1 so that we are left

1 2 12 7 8 with [0, 9 ] ∪ [ 9, 3 ] ∪ [ 3, 9 ] ∪ [ 9 , 1]. Again, each of these intervals is similar to 1 the original interval [0, 1] with the similarity constant 32 and label this new

set F2.

rd k • Step k: Remove the middle 3 from each interval in Fk−1 which leaves 2

1 intervals of length 3k ; this stage is called the kth stage or Fk.

Figure 1.1: The Cantor Set [2]

At each stage, we can count how many intervals there are and fnd their lengths

k 1 which is 2 intervals with length of 3k at the kth stage. It is standard to name each stage of a fractal by F0,F1,F2, ..., Fk where k ∈ N and we will do so regularly in this paper. Note these stages are just a fnite number of representations in the

2 process of construction the Cantor Set and the real Cantor Set is obtained when

k tends to infnity and we denote it as F . We can also take the intersection of all

∞ such Fk to obtain the Cantor Set such as F = ∩k=0Fk.

1 Example 1. Consider the Cantor Set F . Scale F by 3 , then the resulting set is a smaller and yet a similar set to the Cantor Set. This set is an exact copy of the left hand subset of the Cantor Set. See the fgure below.

Figure 1.2: The Cantor Set

Now that we have seen what a fractal could be and how it could be generated, we will now go into the theory of fractals. This theory requires for the reader to recall some basic defnitions and theorems which are stripped bare so only the important material is given.

3 Chapter 2

Basic Defnitions

2.1 Logarithms and Exponential Laws

In this paper we will face many concepts which involve fractals. These fractals are

complex in nature but our focus will be mainly on the dimension of such objects.

Once we give several di˙erent defnitions of dimension we will face how to calculate

these dimensions which involves using techniques of logarithms and exponential

laws. Before we undertake these calculations we need some review.

Defnition 1. If b and x are positive real numbers and y is a real number then,

y y = logb x is equivalent to b = x. We also read this as log base b of x.

In the defnition above, we call the equality y = logb x the logarithmic form and the following equality, by = x, the exponential form. Next we will recall some useful properties of logarithms.

Proposition 1 (Wolfram). Suppose a, b, and c are positive real numbers and p any real number then

4 1. log c ab = log c a + log c b

a 2. log c b = log c a − log c b p 3. logc a = p logc a log a 4. log c a = log c 1 5. logc a = loga c

The above properties can all be verifed by the defnition and properties of exponents.

The following collection of properties which we will call proposition 2 can be proven by the properties in proposition 1. Since the material is review, which the reader has seen, we do not prove these properties but insteadleave them as exercises for the reader.

Proposition 2. (Wolfram)

Suppose a, b, and c are positive real numbers and p is any real number then

1. logc 1 = 0

2. logc c = 1

p 3. logc c = p

4. log c a log a b = log c b

We remind the reader these properties will be helpful when we solving for the

actual dimension of a fractals in the later chapters.

2.2 Topology

Before we establish some conceptsof functions we need some other useful defnitions

from Topology. We will not go into too much theory on the subject but the main

theorem we will like to touch on is the Heine-Borel Theorem.

5 Defnition 2. A topology on a set X is a collection τ of subsets of X having the following properties:

1. ∅ and X are in τ

2. The union of the elements of any sub-collection of τ is in τ

3. The intersection of the elements of any fnite sub-collection of τ is in τ

The set X for which a topology τ has been specifed is called a topological space.

Defnition 3. If X is a topological space with topology τ, then the subset V of X is said to be an open set of X if V belongs to the collection τ.

Defnition 4. A subset A of a topological space X is said to be closed if the set

X − A is open.

Defnition 5. If τ is the collection of all open intervals in the real line,

(a, b) = {x : a < x < b}, the topology generated by τ is called the standard topology on R.

In general, the standard topology on Rn is generated by open balls denoted

B(x, r) = {y ∈ Rn : |x − y| < r} where x is the center of the ball and r is the radius of the ball. We will also recognize other open sets such as open triangles or open

cubes.

Defnition 6. Let X be a topological space. Then the following conditions hold:

1. ∅ and X are closed (and open)

2. Arbitrary intersections of closed sets are closed

6 3. Finite unions of closed sets are closed.

The previous defnition allows for arbitrary unions of open sets to be open.

Similarly, fnite intersections of open sets are open. Both properties are important and very useful when we discuss how to generate fractals.

Now we will go into some more defnitions before ending this section with the

Heine-Borel Theorem. These defnitions are just here for a review for the reader and will be useful when discussing fractals. For the rest of the paper, the topology and space we will be working with will be the standard topology on Rn .

Defnition 7. Let A be a subset of the Rn . The set A is bounded if and only if there exists a non zero M ∈ R such that |x − y| ≤ M for all x, y ∈ A.

Defnition 8. A collection ρ of subsets of a space X is said to cover A, or be a covering for A, if the union of the elements of ρ contain A. It is called an open covering of A if all elements in ρ are open subsets of X.

Defnition 9. A space X is said to be compact if every open covering ρ of X contains a fnite sub-collection that also covers X.

Theorem 1. Heine-Borel Theorem (Munkres) Let A be a subset of topological space Rn . Then A is a compact set if and only if A is closed and bounded.

Example 2. Consider the kth stage of the Cantor Set, Fk. Since Fk is a fnite union of closed intervals it is closed. Since an arbitrary intersection of closed sets is closed T∞ then F = k=0 Fk is a closed set. Since for any x, y ∈ F we know x, y ∈ [0, 1] so |x − y| ≤ 1 Thus, F is bounded. Therefore by the Hiene-Borel Theorem, F is a compact set.

7 2.3 Functions

Although the construction of most fractals can be made with a simple process or

pattern, we often use functions to defne their construction. The use of a function

allows for an analytical approach which is vital when determining the fractals’

structure and dimension. Thus functions play an important role when discussing

how can we create fractals; such functions are called iterated function systems.

Defnition 10. A function f from a set A to a set B is a rule which associates each element in A to an element in B.

The usual notation we use is f : A → B to state that f is a function from A to B.

Defnition 11. Suppose f : A → B. The set {f(x): x ∈ A} is called the range of f or the image under f.

It is good to note here the range of f is a subset of the set B and not necessarily equal to the set B.

Defnition 12. A function f is bounded on a set A if there exists a real positive number M such that |f(x)| ≤ M for all x ∈ A.

We call M the bound of thefunction. Also we say the function f is unbounded if there does not exist such a bound.

Defnition 13. Suppose A is a closed subset of Rn . The function f : A → A is called a contraction if there is a number 0 < c < 1 such that

|f(x) − f(y)| ≤ c|x − y|

for all x, y ∈ A. If the inequality sign is replaced with an equals sign then we call f a similarity.

8 Defnition 14. A function f : A → B is said to be injective (one-to-one) if for each pair of distinct points of A, their image under f are distinct. Now f is said

to be surjective (onto) if every element of B is the image of some element of A under f. If f is both injective and surjective, it is said to be bijective (one-to-one correspondence).

One key concept which comes from this sectionis thata contraction is a bijection and a continuous function.

Proposition 3. A contraction function f : Rn → Rn such that |f(x) − f(y)| ≤

c|x − y| for all x, y ∈ Rn where 0 < c < 1 then f is a bijection and a continuous function.

n Proof. First we will look at continuity, let  > 0 and x1, x2 ∈ R . Let δ = . If

|x1 − x2| < δ then

|f(x1) − f(x2)| = c|x1 − x2| < cδ = c < 

. Thus f is a continuous map.

Next we will show f is injective. Suppose f : Rn → Rn is a contraction such that |f(x) − f(y)| = c|x − y| with 0 < c < 1. Consider g(x) = f(x) + x. Let

n x1, x 2 ∈ R where x1 =6 x2. Assume g(x1) = g(x2) i˙ f(x1) + x1 = f(x2) + x2 i˙

|f(x1) − f(x2)| = |x1 − x2|. This is a contradiction since f is a contraction. Thus f is one-to-one.

For surjective we will show the case when n = 1, that is f : R → R. Let y ∈ R y y then x = c ∈ R since c =6 0. Now |f(x) − f(0)| = c|x − 0| = c| c | = |y|. Since we y y have already shown f is injective we know either c or − c maps to y.

9 Next the following defnition is an alternative form of looking at continuity.

Theorem 2. (Munkres) A function f : Rn → Rn is a continuous function if for any open set U ∈ Rn then f −1(U) is a open set in Rn .

We will not prove this here since there is a nice proof in section 18 page 102 in

Topology by Munkres.

Defnition 15. A family of contraction {f1, f2, ..., fn} where n ≥ 2 is called an iterated function system or IFS.

1 1 2 Example 3. Consider the following set {f1(x)= 3 x, f2(x)= 3x + 3 } where f1, f2 : [0, 1] → [0, 1]. Let x, y ∈ [0, 1]. Then

1 1 |f1(x) − f1(y)| = | 3x − 3 y| 1 = | 3 ||x − y| 1 = 3 |x − y|. and

1 2 1 2 |f2(x) − f2(y)| = |( 3x + 3 ) − ( 3y + 3 )| 1 1 = | 3x − 3 y| 1 = | 3 ||x − y| 1 = 3 |x − y|. 1 Since c = 3 < 1 then we say both f1 and f2 is a contraction and moreover a 1 1 2 similarity. This implies the set {f1(x)= 3 x, f2(x)= 3 x + 3 } is an IFS (iterated function system).

Defnition 16. Let A be a subset of Rn . A compact set F ⊂ A is said tobe strictly

self similar if there exist an IFS {f1, f2, ..., fn} where fi : A → A and the fi’s are similarities for all i and n [ F = fi(F ). i=1

10 Further, F is called the attractor for the IFS {f1, f2, ..., fn}.

Example 4. Consider the unit interval F0 = [0, 1] ⊂ R. Let {f1, f2} be the IFS given in example 3. We know these two functions are similarities. Now if we apply

1 the IFS to each point of [0, 1], we obtain the set F1 = f1([0, 1]) ∪ f2([0, 1]) = [0, 3 ] ∪ 2 1 2 1 2 [ 3 , 1]. Again apply the IFS to F1 to get F2 = f1([0, 3] ∪ [ 3 , 1]) ∪ f2([0, 3] ∪ [ 3 , 1]) = 12 12 7 8 [0, 9 ] ∪ [ 9, 3 ] ∪ [ 3, 9 ] ∪ [ 9 , 1]. As we do this process an infnite number of times, the result will produce the Cantor Set, call it F . Recall F is compact, if we were to

apply the IFS to the Cantor Set F , the result of this would yield another fner copy S2 of itself. Thus F = i=1 fi(F ) and the Cantor Set is strictly self similar. Moreover,

F is the attractor of the IFS {f1, f2}.

n n Defnition 17. Suppose {f1, f2, ..., fn} are similarities where fi : R → R for each 1 ≤ i ≤ n. If there exists an open set U such that

n [ fi(U) ⊂ U i=1 which is a disjoint union then we say the IFS satisfes the open set condition.

Example 5. Consider {f1, f2} in example 3. Now consider the open set U = (0, 1).

1 2 S2 1 We see that f1(U) = (0, 3 ) and f2(U) = ( 3 , 1). Thus i=1 fi((0, 1)) = (0, 3 ) ∪ 2 ( 3 , 1) ⊂ (0, 1). Therefore, {f1, f2} is an IFS which satisfes the open set condition. Furthermore, one can clearly see if we keep iterating to the next step or set that

this new set will still be contained in (0, 1).

The open set condition is a very important property which gives that there is

not too much overlap in a set under iteration. This involves the use of measure

theory to fully explain the open set condition but we will not cover all the details

of the property since we only are worried about fractals and their dimension and

11 not their measure. During this paper we will try to avoid any information which

will stray too far from the subject.

2.4 Fractal Defnitions

As humans, we undertake constructingobjects, such as houses, appliances or motor

vehicles, with one key idea- the idea of perfection. Most of these man-made objects

seek some type of perfection, which the creator envisioned. Also, in nature, some

objects do not seekhuman creativity but instead use natural algorithms to generate

themselves like plants or clouds. In academia, we categorize some of these objects

which are constructed by functions as fractals. Although there are many di˙erent

fractals in the real world and in academia, we will focus our attention to a subset

of fractals called strictly self similar fractals.

Defnition 18. A property of two fgures whose corresponding angles are all equal and whose distances are all increasing by the same ratio are said to be similar.

Example 6. Consider the following two triangles in Figure 1.1. Since the angles

of both triangle ABC and triangle DEF are the same then both triangles are similar

to one another and the ratio of corresponding side lengths are all equal.

Figure 2.1: Similar Triangles [5]

12 Moreover, a similarity transforms fgures to similar fgures.

Although we concern ourselves mainly with the discussion of the dimension of strictly self similar sets, we will introduce some non self similar sets to understand why we need complex defnitions for dimension.

As beings of this Earth, we do not have to venture far to witness perfection frst hand. In nature, many plants and natural objects manifest unique characteristics which seem unnatural but none the less are real. In ferns, the blades follow a unique pattern in which every pinnule appears similar to the pinna. Another example, are the beaches of the world; these beaches may not all have the same size but they appear to have a similar formation which is the shoreline or its sand dunes.

Example 7. Consider the fern and coastline in Figure 2.2 and 2.3. These objects appear to have subsets similar to its larger self and are great examples of how sets may appear to be strictly self similar but in fact are not because not all their subset are exact replicas of the larger whole. For instance, we note the leaf structure of

Figure 2.2. Now each pinnule is not necessarily an exact copy of the pinna. It is easy to see that the top pinna is not similar to the other three as the pinnule structure is di˙erent. Although these objects are not strictly self similar, they are fractals.

The following defnition is just one of many defnitions we run across in mathe- matics. There is not a decisive defnition of what a fractal is but we will show two defnitions so the reader is exposed to at least two forms of defnitions.

Defnition 19. A fractal is an object or quantity that displays self-similarity on

all scales.

Note this defnition does not require strict self similarity on all scales. But if

13 Figure 2.3: A Shoreline [20]

Figure 2.2: Diagram of a Fern [4] we look at one subset it will be self similar to another part.

That is, they appear to have self similar properties but are not considered to be strictly self similar sets which is the case with a coastline or fern found in nature.

There are many interpretations of what a fractal may be defned as and our defnition takes advantage of the characteristic of self similar. Kenneth Falconer’s defnition of a fractal in his book Fractal Geometry (page xxv) is a list of traits to categorize fractals instead of a traditional one sentence classifcation.

Here is a collective list of what a set F may need to posses in order to be classifed as a fractal:

1. F has a fnite structure, i.e. detail on arbitrarily small scales.

2. F is too irregular to be described in traditional geometrical language both locally and globally.

14 3. Often F has some form of self-similarity, perhaps approximate or statistical

4. Usually, the ’fractal dimension’ of F (defned in some way) is greater than its ’topological dimension.’

5. In many cases, particularly the ones investigated here, F is defned in a very simple way, perhaps recursively.

Note item (4) which mentions dimension. We will defne several types of dimen- sions in the subsequent chapters. Benoit Mandelbrot initially defned fractals in terms of dimension in this book, "The Fractal Geometry of Nature." He frst states any subset of Rn has Topological and Hausdor˙ dimension at least zero and at most n. Although Topological dimension is an integer value, Hausdor˙ dimension

may not be, and both dimensions do not have to coincide but only satisfy the tri-

angle inequality. Also for all Euclidean objects (like lines, areas, solids, etc.) both

dimensions will be equal, but for the most part the Hausdor˙ dimension will be

greater than the Topological dimension when considering special sets. So we can

defne these special sets as fractals which have Hausdor˙ dimension exceeding the

Topological dimension. Thus when fractal sets were initially being analyzed they

were defned as having a non-integer (fractional) Hausdor˙ dimension. Mandelbrot

eventually found this simple defnition unsatisfactory for defning fractals and thus

most defnitions are now similar to Falconer’s list. Since dimension is a key in

classifying sets as fractals, we will focus our attention on topic.

2.5 Strictly Self Similar Sets

Mathematical objects created by a similarity process are the easiest fractals to

understand and are some of the most beautiful in the world. For the majority of

15 the paper we will only concern ourselves with strictly self similar sets. We will also

introduce fractals that do not obey the defnition of a strictly self similar set simply

to help us understand why the defnitions are so complex.

2.5.1 The Cantor Set

Earlier in our discussion we gave a process which generated the Cantor Set but now

we introduce a new method which uses an IFS. Recall the following two functions

S1,S2 : [0, 1] → [0, 1] where

1 1 2 S (x) = x; S (x) = x + . 1 3 2 3 3

Then if we apply the functions to each of the points in the unit interval we

1 2 obtain F1 = S1([0, 1]) ∪ S2([0, 1]) = [0, 3 ]∪ [ 3 , 1] which is the frst step in the Cantor

Set. Again we apply the function to the previous set to obtain F2 = S1(F1) ∪

12 12 7 8 S2(F1) = [0, 9 ] ∪ [ 9, 3 ] ∪ [ 3, 9 ] ∪ [ 9 , 1]. Once we do this process k-times, the set 1 2 1 3k−1−1 3k−2 3k−1 Fk = S1(Fk−1) ∪ S2(Fk−1) = [0, 3k ] ∪ [ 3k , 3k−1 ] ∪ ... ∪ [ 3k−1 , 3k ] ∪ [ 3k , 1] where k 1 Fk contains 2 subintervals of length 3 k . Now if we were to let k tend to infnity

then the resulting set would be the Cantor Set F . That is F = limk→∞ Fk or S∞ F = i=1 Fi. Further, recall we know this IFS meets the open set condition.

2.5.2 The Sierpinski Triangle

The next fractal we will discuss is named after Polish mathematician, Waclaw Sier-

pinski (1882-1969), who was well known for his work in Number Theory, Topology

and other areas. This fractal is called the Sierpinski Triangle or Sierpinski Gasket.

The Sierpinski Triangle is defned on R2 . Once again, the manner in which this set is constructed labels itself as a strictly self similar set and involves an infnite

16 Figure 2.4: Waclaw Sierpinski [16]

process. Like the Cantor Set, the Sierpinski Triangle can be generated by a simple

process which follows the following steps:

2 • Step 0: Consider the subset F0 of R where F0 is the equilateral triangle of base 1.

1 • Step 1: Subdivide F0 into four equilateral triangles of base 2 and remove the

middle upside down equilateral triangle from F0 so that there is only left with

1 3 equatorial triangles with base 2 , call this new subset F1.

1 • Step 2: Subdivide F1 into twelve equilateral triangles of base 4 and remove 2 each middle upside down triangle from F1 so that there is only 3 equilateral

1 triangles of base 22 , call this F2.

k k−1 • Step k: Subdivide Fk−1 into 3 + 3 triangles and remove the middle upside

k 1 down triangle from each triangle in Fk−1 leaving 3 triangles with base 2k ,

call this set Fk.

Once again the steps which we lay out for constructing this fractal need to be done an infnite number of times and yields the Sierpinski Triangle. Also lets not

17 Figure 2.5: The Sierpinski Triangle [15]

T∞ forget that Fk ⊂ Fk−1 ⊂ ... ⊂ F1 ⊂ F0 and k=1 Fk = F where F is said to be the Sierpinski Triangle.

The second method follows a more algebraic approach which involves the use

of an iterated function system. Consider the following three functions S1,S2,S3 :

R2 → R2 where √ 1 1 1 1 1 1 1 1 3 S (x, y) = ( x, y); S (x, y) = ( x + , y); S (x, y) = ( x + , y + ). 1 2 2 2 2 2 2 3 2 4 2 4

We can do the same process to iterate the Sierpinski Triangle as we did with

2 the Cantor Set. Consider the closed set F0 ⊂ R where F0 is the equatorial triangle of base 1. Now apply the IFS to F0 so that F1 = S1(F0) ∪ S2(F0) ∪ S3(F0). This iteration obtains the frst stage in the Sierpinski Triangle. If we apply the IFS k-times then we obtain the following equation Fk = S1(Fk−1) ∪ S2(Fk−1) ∪ S3(Fk−1) √ k 3 where Fk contains 3 sub triangles of area 4k+1 . If we were to let k tend to infnity the outcome would be the Sierpinski Triangle F . S3 Since F = limk→∞ Fk we know F = i=1 Si(F ) thus F is a strictly self similar T∞ set. Since each Fk is the fnite union of closed sets, it is closed and so F = k=1 Fk

18 is also closed. Furthermore, F is clearly bounded so F is compact. Thus F is a strictly self similar set and the attractor of the IFS.

The IFS for the Sierpinski Triangle satisfes the open set condition as seen in

the following example.

Example 8. Consider the open set V which is the open equatorial triangle of base

one. Let {S1,S2,S3} be the IFS, as seen above, which generates the Sierpinski Triangle. Then if we apply the IFS to V then we get a smaller set which is still contained in V . Ultimately, any number of iterations of V under the IFS are subsets of V . Thus {S1,S2,S3} satisfy the open set condition. See Figure 2.5.

Figure 2.6: Open set under IFS

2.5.3 The Von Koch Curve

The next fractal we will introduce is the beautiful Von Koch Curve. This fractal

was frst mentioned in a paper in 1904 by Helge Von Koch (1870-1924) and is one

of the earliest fractals to be classifed. To produce this set we have written out

some steps to show how this fractal is generated. After we show how to generate

this fractal with step by step process then we will show how an IFS can generate

19 this fractal. Thus in doing so we can show the Von Koch Curve is also a strictly

self similar fractal.

2 • Step 0: Consider the subset F0 of R where F0 is the line segment [0, 1].

1 • Step 1: Subdivide F0 into three equ a l line segments of length 3 and remove

the middle third line segment from F0 so that we are only left with two line

1 2 1 segments [0, 3] ∪ [ 3 , 1]. Now take two line segments of length 3 and connect them at one endpoint. Next place one endpoint of the jointed line segments

1 at the right endpoint of [0, 3 ] and the endpoint of the segment attach to the 2 left end point of [ 3 , 1], call this new set F1. Note there are 4 line segment in this set. See Figure 2.7.

1 • Step 2: Subdivide F1 into twelve line segments of length 32 and remove the middle third from each line segment. Next take four pairs of line segments

which are joint together at one of their endpoints. Attach these pairs at the

endpoints of were the middle third piece were removed, call this set F2. Note there are 42 line segments in this set.

k k−1 • Step k: Subdivide Fk−1 into 3 + 3 triangles and remove the middle upside

k 1 down triangle from each triangle in Fk−1 leaving 3 triangles with base 2k .

Moreover the step by step process is an infnite process which constructs the

Von Koch Curve.

Now for the second method we can construct the Von Koch Curve with the use

2 2 of an iterated function system. Consider the four functions S1,S2,S3,S4 : R → R where

20 Figure 2.7: The Von Koch Curve Figure 2.8: Helge Von Koch [11] [7]

⎛ ⎞ ⎛√ ⎞ ⎛ ⎞ ⎛ ⎞ 1 − 3 1 1 x 1 2 2 x 3 S1(x, y) = ⎜ ⎟ ; S2(x, y) = ⎜ √ ⎟ ⎜ ⎟ + ⎜ ⎟ ; 3 ⎝ ⎠ 3 ⎝ 3 1 ⎠ ⎝ ⎠ ⎝ ⎠ y 2 2 y 0

⎛ √ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 1 3 1 2 1 2 2 x 2 1 x 3 S3(x, y) = ⎜ √ ⎟ ⎜ ⎟ + ⎜ √ ⎟ ; S4(x, y) = ⎜ ⎟ + ⎜ ⎟ . 3 ⎝ − 3 1 ⎠ ⎝ ⎠ ⎝ 3 ⎠ 3 ⎝ ⎠ ⎝ ⎠ 2 2 y 6 y 0 Now we will show these functions can generate the Von Koch Curve. Consider

1 the closed interval F0 = [0.1]. F1 = S1(F0) ∪ S2(F0) ∪ S3(F0) ∪ S4(F0) = [0, 3 ] ∪ √ √ √ √ √ 1 − 3 1 3 1 1 3 1 − 3 1 3 {( 6 x + 6 y + 3, 6 x + 6 y):(x, y) ∈ F0 } ∪ {( 6x + 6y + 2, 6 x + 6y + 6 ): 1 (x, y) ∈ F0} ∪ [ 3 , 1]. Now if we do this k-times then we will get the set Fk = 1 1 1 S1(Fk−1) ∪ S2(Fk−1) ∪ S3(Fk−1) ∪ S4(Fk−1) = {( 3x, 3 y):(x, y) ∈ Fk−1} ∪ {( 6 x + √ √ √ √ √ − 3 1 3 1 1 3 1 − 3 1 3 6 y + 3, 6x + 6 y):(x, y) ∈ Fk−1} ∪ {( 6x + 6y + 2, 6 x + 6 y + 6 ):(x, y) ∈ 1 2 1 Fk−1} ∪ {( 3 x + 3, 3 y):(x, y) ∈ Fk−1 }. If we were to let k go to infnity then the outcome would be the Von Koch Curve.

2 S4 If F is the Von Koch Curve, F = limk→∞ Fk in R . Thus F = i=1 Si(F ) and so F is a strictly self similar set. Since the union of arbitrary closed sets is not

21 necessarily closed. Since each Si is a contraction it is a bijection and continuous.

2 S4 Since F0 is closed in R Si(F0) is closed and the i=1 Si(F0) is closed. Repeating this process as k → ∞ gives F is closed. F is clearlybounded in the unit triangle which implies F is compact. Therefore F is a strictly self similar set and the attractor of the IFS.

Moreover, like the Sierpinski Triangle, the Von Koch Curve also satisfes the

open set condition. Since this set is a subset of R2 then we must choose an open set in R2 . (i.e. (0,1) is not open in R2) So instead we choose the open trapezoid with √ 1 3 0 0 bases 1 and 2 , height of 4 , and angles of 60 and 120 as shown in the following example.

Example 9. Consider the open trapezoid V described above. Let {S1,S2,S3,S4} be the IFS which generates the Von Koch Curve. Apply the IFS to V which yields a

V1 ⊂ V . Applying the IFS k-times will generate Vk ⊂ V . Therefore, {S1,S2,S3,S4} satisfes the open set condition. See Figure 2.8.

Figure 2.9: Open Trapezoid under IFS

22 Chapter 3

Non-Similar Fractals

3.1 The Julia Sets

Although the reader was introduce to strictly self similar sets, we will now mention how the Julia sets do not ft the criteria of what it means to be a strictly self similar set. The mathematician who found one of the earliest non self similar sets was Gaston Maurice Julia (1893-

1978). During World War I he fought the Germans in which he lost his nose. He did in fact receive an award for this bravery called the Legion of Honour. In spite Gaston Maurice Julia [8] of his disfgurement, he was able to witness some of the most amazing fractals which come from a relatively simple looking transformation.

This transformation is f : C → C, where given c ∈ C, is defned as:

f(z) = z 2 + c.

The set themselves are determined by the choice of c. For a given value of C

23 k the the Julia set, Jc, is z ∈ C such that f (z) is bounded for all k ∈ N. There are many di˙erent sets which come from this process, many of which are fractals, but we will only show two such sets that will dazzle the readers eyes.

Example 10. Consider the complex number c = −0.122116107 − 0.748590931i.

Defne the transformation f(z) = z2 + c over the complex plane. Therefore the image below is the boundary of all points that converge to a value (less than infnity)

after repeated iteration under f.

Example 11. Consider the complex number c = −1.310092059 + 0.012130511i.

Defne the transformation f(z) = z2 + c over the complex plane. Again the image below is the boundary of all points that converge to a value (less than infnity) after

repeated iteration under f.

These sets appear to have similarity in some segments. Indeed we see some symmetry. But we cannot take any part of this set which gives us back the whole fractal. Thus these fractals are not strictly self similar.

24 3.2 The Mandelbrot Set

For most of my life, one of the persons most ba˜ed by my own work was myself.

Benoit Mandelbrot

The Mandelbrot Set is one of the most interesting sets we will see in our life time. This set is not a strictly self similar set and in this section we show one argument why. The Mandelbrot Set is named after Benoit Man- delbrot (1924-2010) who frst used a computer, in 1979, Benoit Mandelbrot [9] to visually depict this alluring set.

2 Consider the following sequence of complex numbers zn+1 = zn + c where c ∈ C.

We defne the Mandelbrot Set as M = {c : supn→∞|zn+1| ≤ 2, z0 = 0, zn+1 = zn +c}.

Figure 3.1: The Mandelbrot Set [17]

The Mandelbrot Set is generally all numbers c such that zn+1 is bounded and

does not tend to infnity, where z0 = 0. The fgure above classifes the Mandelbrot Set as the black region in the center and all the other points with di˙erent shades

25 of gray which labels them respectively how fast they tend to infnity. We will

demonstrate how a point inside the set is bounded and a point outside will jump

to infnity in the next two examples.

Example 12. Consider c = 1. Then consider the following sequence of complex

2 numbers zn+1 = zn + 1. Let z0 = 0. Thus the sequence is 0, 1, 2, 5, 26, ... which is a unbounded sequence of numbers. Therefore, 1 6∈ M.

1 1 Example 13. Consider c = − 4 + 4 i. Then the sequence associated with c is 2 1 1 1 1 1 zn+1 = zn + − 4+ 4 i. Let z0 = 0. Then the sequence of numbers is 0, − 4+ 4i, 4 + 3i 11 7 361 205 1 1 8, 64 + 16i, 4096 + 512 i, ... which converges. Therefore, − 4 + 4 i ∈ M

Before we show why the Mandelbrot set is a not a strictly self similarset there are a few defnition which we need to address. These following defnitions characterize the Mandelbrot set in a manner which will lead up to the relationship between the Fibonacci numbers and the period of the primary bulbs. This relationship will allow us to fnd the radius of each bulb and relate this information to justify why the Mandelbrot set is not a strictly self similar set.

Defnition 20. The main cardioid is the largest region in the Mandelbrot Set or

the heart-shaped fgure.

Defnition 21. A primary bulb is a disk shaped object connected to the main cardioid.

Defnition 22. For each primary bulb there are infnity many antennas (string like attachments), the largest antenna is called the main antenna.

Defnition 23. The period of a bulb is a natural number, n, which is associated

2π to the angle where the center of the bulb is located, n .

26 Defnition 24. The number of spokes in the main antenna is the period of the

bulb.

p Defnition 25. The rotation number of a bulb is a rational number q where The denominator is simply the period of the bulb (the number of spokes) and numerator

is the position of the smallest spoke in a counterclockwise direction.

Figure 3.2: Main Cardioid, Primary bulbs, Spokes, and Main antennas [6]

Another intuitive way to look at the Mandelbrot set is by thinking of the main

1 cardioid as a circle of radius 2 . The fgure below makes it easier to understand the following defnition of period. Also it is helpful to recall, the period is the least

positive value required for the function to make one complete full cycle.

27 th 1 Example 14. Consider the 4 period bulb. This rotation number of this bulb is 4 since the smallest antenna is in frst position and period of the primary bulb is 4.

Figure 3.3: 4th primary bulb on Figure 3.4: Smallest antenna on the Mandelbrot Cactus [6] the 4th primary bulb [6]

A fact about the periods of the primary bulbs is that they can be associated to the Fibonacci numbers. This relationship can be witnessed when we label the largest bulb in the Mandelbrot Cactus 1 and the left half bulb 2. These numbers are choose not by ordering the bulbs by size but by their period. Next we fnd the largest bulb between bulbs 1 and 2 which is a bulb of period 3. Again take

both bulbs of period 2 and 3 and fnd the largest bulb in between them which is a bulb of period 5. Do this process in an orderly fashion and the outcome will be the

Fibonacci numbers as seen in the fgure.

Now we go back to the question ask in beginning of this exploration of the

Mandelbrot set, "Is the Mandelbrot set a strictly self similar set?" Well the answer is no. The reason is because the antennas at each bulb is changing as we zoom in more and more. The antennas are associated to the Fibonacci numbers which have

28 Figure 3.5: Fibonacci Numbers Found in the Mandelbrot set [6]

no common scaling factor. Yes for every nth period bulb there are n − 1 copies of an nth bulb which all have a di˙erent rotation number. Thus, each bulb "appears" to look similar to the other bulbs, or sets as a whole, but is not strictly self similar.

29 Chapter 4

Dimension

The dimension we were taught in high school is not the only way to measure dimen-

sion. For most of us, we thought of dimension as whole numbers such as assigning

a dimension of 1 to a line or a solid object having dimension 3. In mathematics, the dimension of a space is the minimum number of parameters needed to specify any point in the given space. Nonetheless, we are taught an object must have a whole number dimension. However this concept of dimension does not explain the di˙erence between a square and the Sierpinski Triangle. Yes, length, area, and volume are related to the dimension of many objects but how can we determine the dimension of complex sets like the Cantor Set or the Sierpinski Triangle, or a coastline. Especially in nature, not all objects are strictly self similar like the coastline. Therefore it is useful to have tools like dimension to determine the struc- ture of an object in our physical world. These are the questions generally asked by mathematicians when they talk about dimension and which we will address in the following sections.

30 4.1 Topological Dimension

When considering a set A in Rn this set is related to a real number which is the

set’s dimension. The one property any subset A of Rn is that the dimension of A cannot exceed n nor be less than 0. The number which we speak of can be any real number but when we consider Topological dimension we only consider the subset

A to have an integer from 0 to n as its dimension. The following two defnition can be found in Munkres’ book Topology on page

305 section 50. This defnition is diÿcult to understand but state the topological

n dimension of A , a set in R , is always an integer, dimT (A) = m for some integer

m ≤ n. If A is totally disconnected dimT (A) = 0 and if each point in A has an

arbitrarily small neighborhood with boundary of dimension 0 then dimT (A) = 1 and so on.

Defnition 26. A collection A of subsets of A is said to have order m + 1 if some

point of A lies in m + 1 elements of A, and no point of A lies in more that m + 1 elemetns of A.

Defnition 27. A set A is said to be fnite dimensional if there is some integer m such that for every open covering A of A, there is an open covering B of A that refnes A and has order at most m + 1. The topological dimesion of a set A is defned to be the smallest value of m for which this statements hold; we denote it by dimT A.

Now this defnition may seem fne at frst when we talk about the world we live in but this defnition is quite fawed. There are many sets which will not sit nicely with this defnition of dimension. Among these sets, the fractals, have much space added, or missing, from them than other forms of sets with the same topologies.

31 4.2 Similarity Dimension

In our study of fractals, we mentioned their that are two types of these geometric

objects which are the strictly self similar sets and fractals that are not strictly self

similar. When we work with fractals that are strictly self similar then typically we

calculate their dimension using the Similarity Dimension.

A direct way of fnding the dimension of a strictly self similar set is looking at

a relationship between the set and the smaller scaled subsets. We frst start with

the following equation 1 n = ( )d c where n is the number of subsets which are similar to the whole set and c is the scaling factor of the subsets.

Next take the log of both sides

1 d log n = log c

1 −1 Then manipulating the exponent of c = c and using the power rule of logarithms we get

log n = −d log c or log n − = d log c where 0 < c < 1. Now we have established enough information to properly defne the Similarity Dimension.

Defnition 28. Given a strictly self similar set F , with IFS {S1,S2, ..., Sn} and similarity constant c then 1 n = ( )d . c

32 or more intuitively log n dim (F ) = − = d S log c where d is called the Similarity Dimension of F .

Example 15. Consider the unit cube I. If we were to scale the cube by a third

then we would need 27 third sized cubes in order to construct back the original cube.

Therefore, if n is the number piece needed to produce the original set and c is the scaling factor of the smaller subsets. Then

1 d 27 = ( 1 ) 3 27 = 3d .

Figure 4.1: Unit Cube [1]

This implies the Similarity Dimension of the cube is 3 which is true since the cube is a 3-dimensional object. Another fact we will point out is that Similarity

Dimension of a cube agrees with its topological dimension, the dimension we thought

we understood in high school.

Example 16. Consider the unit interval [0, 1]. Suppose we cut the unit interval in

1 1 half such as [0, 1] = [0, 2 ] ∪ [ 2 , 1]. Then the number pieces the interval broken up 1 into is 2 where both subintervals have a scaling factor of 2 . Therefore

log 2 log 2 dimS ([0, 1]) = − 1 = = 1. log 2 log 2

33 Example 17. Consider the Cantor Set F . Then we know there exists similarities

1 {S1,S2} with scaling factor 3 to create the Cantor Set thus:

log 2 log 2 dimS (F ) = − 1 = . log 3 log 3

log 2 Therefore, the Similarity Dimension of the Cantor Set is log 3 .

It is worth noting that for a strictly self similar set with similarity dimension

Pn d 1 d d d then i=1 c = 1. Simple calculation show if n = ( c ) then nc = 1 thus

Pn d i=1 c = 1. This fact will be used in the subsequent chapters. We will also use

d 1 c = n .

If F is the attractor for an IFS {S1,S2, ..., Sn} with |Si(x) − Si(y)| = c|x − y| for all i then the Similarity dimension d of F is the unique number such that

n X c d = 1. i=1 This is easily shown with logarithums since

n X c d= nc d i=1

d 1 log n and nc = 1 i˙ n = cd i˙ log n = −d log c i˙ − log c = d and by defnition log n dimS (F ) = − log c . Similarity dimension is only defned for strictly self similar sets which is a subset of fractals. There are many other defnitions of dimensions such as the Hausdor˙

Dimension which is used to fnd the dimension of a large class of sets other than the strictly self similar fractals. Initially, fractals were defned in terms of the Hausdor˙ dimension.

34 4.3 Hausdor˙ Dimension

Now looking back at all the di˙erent fractals we know, many of them are not strictly

self similar and so cannot be calculated by the Similarity Dimension. This is why

we use Hausdor˙ Dimension to describe fractals. Calculating Hausdor˙ dimension

is much more diÿcult than calculating Similarity Dimension. We will do our best

to help the reader better understand this type of dimension and its calculation. We

will give simple examples so the concept of Hausdor˙ Dimension is clarifed.

4.3.1 Hausdor˙ Measure

Defnition 29. Suppose U is a non-empty subset of Rn, for some positive integer n, the diameter of U, denoted as |U|, and defned as

|U| = sup{|x − y| : x, y ∈ U}.

This diameter is the least upper bound of the distance between any two point in the set U.

Example 18. Consider the subset [0, 1] of R. Then |[0, 1]| = sup{|x − y| : x, y ∈ [0, 1]} = |1 − 0| = 1.

Example 19. Consider the Sierpinski Triangle F . Notice at every iteration this set is losing area but the vertices of the original triangle remain in the set. Since any other two points would give a smaller distance than the original three points in the frst stage of F then |F | = |(0, 1) − (0, 0)| = 1. See Figure 4.2.

Defnition 30. We say the countable collection {Ui} is a δ-cover of F S∞ if F ⊂ i=1 Ui and 0 < |Ui| ≤ δ for all i.

35 Figure 4.2: The Sierpinski Triangle

Note that a δ-cover contains the original set but we are not guaranteed an

equality.

We mention briefy there are many types of δ-covers that we can use when

covering a space. Since δ is an arbitrary real positive number then there can be many di˙erent covers which will satisfy being a δ-cover. Namely the covers which

have a diameter smaller than or equal to δ.

Example 20. Consider [0, 1] ⊂ R. Consider the following countable sets which

3 −1 1 1 2 5 17 1 contain [0, 1]: {A1} = {[0, 1]}, {Bi}i=1 = {( 2, 8), ( 9, 3), ( 8, 16 )}, {Cn} = {[− n , 1+ 1 n −1 1 1 1 2 1 2 1 1 n ]: n ∈ N}, {D}i = {( n , 3 +n ), ( 3, 3 +n ), ( 3 − n, 1+ n ): n ∈ N}. Since |A1| = 1, −1 1 1 2 5 17 5 1 1 n+2 |Bi| = sup{|( 2, 8)|, |( 9, 3)|, |( 8, 16 )|} = 8 , sup |Cn| = sup |1 + n− (− n )| = | n | = n 1 2 1 1 1 2 1 2 7 3, and |Di | = sup{ 3 + n , 3 + n , 3 + n } = sup{ 3 + n } = 3 . Then all these are δ−cover of [0, 1] when δ = 3.

What we see is that as δ becomes smaller there fewer sets that are δ−covers.

5 n Now if we were to let δ = 8 then the only sets which would cover [0, 1] are {D}i

36 3 when n ≥ 7 and {Bi}i=1.

Defnition 31. Suppose F is a subset of Rn and s is any positive real number. For any δ > 0 defne ( ∞ ) s X s H δ (F ) = inf |Ui| : {Ui} is a δ − cover of F . i=1

s s Where limδ→0 Hδ (F ) = H (F ) is called the s-dimensional Hausdor˙ measure of F.

This defnition may be diÿcult concept to understand but all it really means is

sum up all the elements in each δ-cover raised to the sth power and choose from all the δ-covers which makes the sum minimal. Like we saw in the previous example we can also state that as δ → 0 then the amount of δ-covers will decrease. Thus

s Hδ (F ) will increase since as δ gets smaller since we have fewer covers from which to obtain the infnitum. That is when we "throw out" sets, the lower bound cannot get smaller but it could stay the same or increase. Furthermore, as δ → 0 we will approach a limit, possibly positive infnity. Now once we calculate the Hausdor˙ measure of subset F of Rn we will see that we obtain a measure of zero or infnity. To see this consider the following example.

2 1 Example 21. Consider the boundary of the unit square F in R . Let B(pi, 2n ) 1 be the closed ball centered at the boundary point pi with radius 2n . See Figure 4.3. 1 Now the diameter of each closed balls is n and since the boundary of the square involves four sides then we can initially cover the square with 4n of these balls.

1 Furthermore, for any δ there exists an n for which {B(pi, 2n )} is a δ−cover. Thus we have for any δ:

4n 4n s X 1 s X 1s 4n H (F ) = |B(pi, )| = | | = . δ 2n n ns i=1 i=1

37 Figure 4.3: Closed balls covering boundary of unit square.

Now as we let δ → 0 then n → ∞ since the individual coverings must become smaller. Moreover, the radii of the closed balls become smaller and thus we need an infnite number of closed balls to cover the boundary of the square.

4n 4n Therefore if s > 1 then limn→∞ ns = 0. If s < 1 then limn→∞ ns = ∞. Therefore s s s when s > 1 then H (F ) = limδ→0 Hδ (F ) = 0; when s = 1 then H (F ) = 4; when s ≤ 1 then H s = ∞.

Theorem 3 (Falconer). Let F ⊂ Rn and f : F → Rn be a mapping such that

|f(x) − f(y)| = c|x − y| (x, y ∈ F ) for constants c ∈ R, 0

H s(f(F )) = cs H s(F ).

Proof. Let {Ui} be a δ−cover of F then f(Ui) is a cδ-cover of f(F ). Now |f(Ui)| =

s P∞ s c|Ui|. Thus, H cδ (f(F )) = inf{ i=1 |f(Ui)| : {f(Ui)} is a cδ − cover of f(F )} = P∞ s s s P∞ s inf{ i=1 c |Ui| : {Ui} is a δ − cover of F } = inf{ c i=1 |Ui| : {Ui} is a δ −

38 s s s cover of F } = c H δ (F ). Now as δ → 0 we know cδ → 0, thus H (f(F )) = csH s(F ).

Now we are ready to fully defne the Hausdor˙ Dimension.

Defnition 32. Let F ⊂ Rn and s is any positive real number then we defne

s s dimH F = inf{s : H (F ) = 0} = sup{s : H (F ) = ∞}.

We will denote dimH F = s as the Hausdor˙ dimension of F.

Figure 4.4: The Hausdor˙ dimension is the value where jump discontinuity occurs

[3]

Again to shed light on this defnition we will go over more examples so the

concept of Hausdor˙ dimension seems understandable and show there is more than

one way to calculate the dimension of sets.

1 Example 22. Consider the unit interval [0, 1]. Let {B(pi, 2n )} be a δ−cover of 1 [0, 1] where B(pi, 2n ) are closed intervals centered at some point pi ∈ [0, 1] of length 1 n . Since the diameter of each closed intervals is its length then

39 n n s X 1 s X 1 s n H ([0, 1]) = |B(pi, )| = | | = . δ 2n n ns i=1 i=1 If δ → 0 then n → ∞ since the individual coverings become much smaller.

1 Moreover, the length of the B(pi, 2n ) are becoming smaller and so we need more 1 n B(pi, 2n ) to cover the unit interval. Thus if s > 1 then limn→∞ ns = 0. If s < 1 n then limn→∞ ns = ∞. Therefore, dimH ([0, 1]) = 1.

2 Example 23. Consider the unit square F of R . Let Ui be closed squares centered

1 at some point pi ∈ F with side-length n . Now the diameter of each closed square is √ 2 n then the Hausdor˙ measure is given by the following summation:

2 2 √ √ n n 2 s s X s X 2 s n 2 H (F ) = |Ui| = | | = . δ n ns i=1 i=1 Now as we let δ → 0 then n → ∞ since the individual coverings must become

√ s √ s n2 2 n2 2 smaller. Thus if s > 2 then limn→∞ ns = 0. If s < 2 then limn→∞ ns = ∞.

Therefore, dimH (F ) = 2.

Example 24. Consider the Cantor Set C. When we discussed the Cantor Set in previous sections the Cantor Set broke up into two pieces every iteration, mainly

1 2 left and right. (i.e. CR = C ∩ [0, 3 ] and CL = C ∩ [ 3 , 1]). Now we also stated that 1 s 3 Ck is similar to both CR and CL where k ∈ N. Assume the H (C) > 0 and s is any positive real number. Since both left and right piece are disjoint we can account

the measure of C as two separate pieces and thus by the Scaling Property,

s s s s 1s 1 1 s s 1 s s H (C) = H (CL)+H (CR) = H ( C)+H ( C) = ( ) H (C)+( ) H (C). 3 3 3 3

s 1 s s s Which simplifes to the equation H (C) = 2( 3 ) H (C). If we divide by H (C) 1 s we get the following equation 1 = 2( 3 ) and using logarithm properties we obtain log 2 s = log 3 which is the Hausdor˙ dimension of the Cantor Set.

40 Example 25. Consider the Von Koch Curve F . Since F is generated by iterating

S4 4 an IFS an infnite number of times then F = i=1 Si(F ) where {Si}i=1 is the IFS for the Von Koch Curve. Assume H s(F ) > 0. By Theorem 3 and by measure theory which we will go into some detail in the following chapter (i.e. see defnition

33) then

4 4 4 4 s s [X s X s 1X 1 s s H (F ) = H ( Si(F )) = H (Si(F )) = H ( F ) = () H (F ). 3 3 i=1 i=1 i=1 i=1

In fact the Si(F ) do not overlap in more than a countable number of points. For now, we take this property for granted and move forward. So by scaling prop-

s P4 1 s s erty. Consider the following equation, H (F ) = i=1( 3 ) H (F ). If we divide s log 4 by H (F ) and solve for s by logarithm properties we get s = log 3 which is the dimension of the Von Koch Curve.

41 Chapter 5

Comparing Dimension

We saw from the examples in the previous chapter that calculating the Hausdor˙

dimension for strictly self similar sets gave us their Similarity dimension. In this

chapter we will show how both Hausdor˙ and Similarity dimension are always equal

on a strictly self similar set. This is not an easy exercise and we will break the proof

into two parts. We will frst show the lower bound of the inequality and then the

upper bound.

Defnition 33. We defne µ, a measure on Rn, if µ assigns a non-negative real number, possibly ∞, to each subset of Rn such that:

1. µ(∅) = 0

2. µ(A) ≤ µ(B) if A ⊂ B

3. A1,A2,A3, ... is a countable (or fnite) collection of sets then

[X µ( Ai) ≤ µ(Ai) i i SP where µ( i Ai) = i µ(Ai) if the Ai are disjoint Borel sets.

42 The classifcation of Borel sets is the smallest collection of subset of Rn that includes:

1. Every open set

2. Every closed set

3. Every countable or fnite union of Borel sets

4. Every countable or fnite intersection of Borel sets

Defnition 34. We defne µ, a mass distribution on Rn, if µ is a measure and

µ(Rn) is fnite.

Example 26. Let α be a point in Rn and defne µ for any set A in Rn be µ(A) = 1 if α ∈ A and µ(A) = 0 otherwise. Clearly, µ meets the frst two parts of the measure defnition and also the inequality in the third part. As to the equality of the third part, if the Ai are disjoint Borel sets we consider the following cases. First

if α ∈ Ai for some i then µ(Ai) = 1 and µ(Aj ) = 0 for i = 6 j since the sets are S P disjoint and thus α 6∈ Aj . This gives µ( i Ai) = 1 = i µ(Ai) so the condition is SP met. Secondly, if α 6∈ Ai for all i then clearly we have µ( i Ai) = i µ(Ai) = 0.

n Theorem 4 (Falconer). Let S1,S2, ..., Sm be contractions on D ⊂ R so that

|Si(x) − Si(y)| ≤ c|x − y| for any x, y ∈ D with c < 1. Then there exists a unique non-empty compact set F that is invariant for the Si, i.e. which satisfes m [ F = Si(F ) i=1

43 Moreover, if we defne a transformation S on the class ρ of non-empty compact sets by ∞ [ S(E) = Si(E) i=1 and write Sk for the kth iterate of S given by S0(E) = E, Sk(E) = S(Sk−1(E)) for

k ≥ 1 then ∞ \ F = Sk(E) k=1

for any set E in ρ such that Si(E) ⊂ E for each i.

We will not prove this theorem since many topics which would be necessary are

not covered in this paper. The key thing which comes from this theorem which is

k n useful for our purposes is the the fact that F = limk→∞ S (E) in R . For the rest of the paper, we will be proving an important theorem of how

Similarity and Hausdor˙ dimension relate. First we will show the lower bound of the argument which is the more interesting of the two.

m Let {S1,S2, ..., Sk} be an IFS of similarities on R where |Si(x)−Si(y)| = c|x−y| for all x, y ∈ Rm, 0

0 k k−1 of S given by S (V ) = V , S (V ) = S(S (V )) for k ≥ 1. Further, let E0 = V ,

Sk n−1 E1 = i=1 Si(V ),..., En = S (V ). As we have previously seen, each Ei is a

n n collection of disjoint sets and Ei ⊂ Ei−1. We defne Vm, 1 ≤ m ≤ k , as one of

n 0 Sk 1 Sk the k sets in En. Note, this gives E0 = V = V 1 , E1 = i=1 V i = i=1 Si(E0), Sk2 2 Sk Skn n n−1 Sk E2 = i=1 Vi = S( i=1 Si(E1)),..., En = i=1 Vi = S ( i=1 Si(En−1)). For example, see Figure 5.1 which is iterated sets from the Von Koch Curve’s IFS on the open trapezoid.

44 n Figure 5.1: The sets En = Vm where n = 0, 1, 2

m 0 Now we defne µ for any set A in R as follows: µ(E0) = µ(V ) = µ(V1 ) = 1

n ns n log k and µ(V m ) = c where n = 1, 2, 3, ... and 1 ≤ m ≤ k and s = − log c . s −1 i i We note here that c = k , as shown on page 34, and that |V j | = c assuming that |V | = 1. Further, we recall from the defnition of similarity dimension that

Pk s i=1 c = 1. And for any set A in Rm we defne:

X i ¯ [ ¯ i µ(A) = inf{ µ(Vj ): A ∩ V ⊂ V j }. i,j i,j

¯ i Thus, if A is any set such that A∩V can be covered by Vj , we choose a covering i ¯ to minimize the sum of the measures of the Vj . We note that if A∩V = ∅, µ(A) = 0; if A ∩ V¯ =6 ∅, then 0 < µ(A) since V¯ covers A ∩ V¯ so there is at least one covering of the intersection.

Example 27. Consider the open trapezoid V from section 2.5.3 and IFS which generates the Von Koch Curve and the sets A and B given in Figure 5.2. Let

2 S4 1 C = A ∪ B ⊂ R . Since C ∩ V 6⊂ j=1 V j then µ(C) = µ(V ) = 1. Similarly,

45 S4 1 S16 2 A ∩ V 6⊂ j=1 V j so µ(A) = µ(V ) = 1. Since B ∩ V 6⊂ i=1 V i then µ(B) =

1 1 µ(V2 ) = 3 .

i Figure 5.2: C ∩ Vj where i = 0, 1, 2

Now, µ is NOT a measure and mass distribution which can be easily verifed and shown below. Now we know µ(∅) = 0. If A ⊂ B then A ∩ V ⊂ B ∩ V . Thus, for any cover C such that B ∩ V ⊂ C we have that A ∩ V ⊂ C. Since we take the infmum over all possible coverings we have µ(A) ≤ µ(B). Also, if µ(V ) = 1 then

µ(R) = 1 since V ∩ R = V .

Further, if A adn B are sets and µ(A) = a, µ(B) = b, and if we let UA be the corresponding covering of A ∩ V and UB is the correspoding cover of B ∩ V . Then any covering of (A ∪ B) ∩ V is contained in UA ∪ UB and so we have µ(A ∪ B) ≤

µ(A) + µ(B). By induction, we can conclude that if A1,A2, ..., An, ... is a countable

46 (or fnite) collection of sets then

∞ ∞ [ X µ( Ai) ≤ µ(Ai). i=1 i=1 However, µ is not a measure since we do not get a equality for the Borel sets. (i.e.defnition item (iii) fails) For example, consider the Von Koch Curve’s open tranpeziod V and dividing this open set into two disjoint sets, call them A and

B.Then µ(A) = µ(B) = 1. Thus, the sum of the measure is 2 but µ(A ∪ B) =

µ(V ) = 1 ≤ µ(A) + µ(B) = 2.

Although µ is not a measure, it will still be very helpful in obtaining the lower

bound for the Hausdor˙ dimension of a strictly self similar fractal F .

m Further, for this IFS {S1,S,2 ..., Sk} on R with the open set condition and

|Si(x)−Si(y)| = c|x−y|, since each Si is a similarity, then there is a unique attractor F such that the Similarity Dimension, s, of F satisfes kcs = 1. We can use µ to

P i ¯ S ¯ i get a lower bound of dimH (F ). So µ(F ) = inf{ i,j µ(Vj ): FV ∩ ⊂ i,j V j }. Since Sn(V¯ ) converges by Theorem 4 to F we know there exists an n such that for all

i i Sk ¯ i Sk ¯i ¯ i > n, F ⊂ j=1 Vj and F = limi→∞ j=1 Vj . Thus, F ∩ V 6= ∅ and 0 < µ(F ). Now,

ki ki ki [ ¯ i X si X ¯ i s 0 < µ(F ) ≤ µ( V j )= c = |Vj | (5.1) j=1 j=1 j=1 i is since |V j | = c for all i. Further, ki ki ki ki X i [X¯ i si [ ¯ i X ¯i s [ ¯ i inf{ µ(Vj ): F ⊂ V j } = inf{ c : F ⊂ Vj } = inf{ |Vj | : F ⊂ V j }. i,j i,j j=1 j=1 j=1 j=1 Recall,

ki ki ∞ ki ∞ s [ ¯ i s [ ¯ i X s [ ¯ i [ H ( Vj ) = lim H ci ( V j ) = lim (inf{ |Ui| : V j ⊂ Ui}). (5.2) ci→0 ci →0 j=1 j=1 i=1 j=1 i=1

47 Clearly,

∞ ki ∞ ki X s [ ¯ i [ X ¯i s inf{ |Ui| : V j ⊂ Ui} = |Vj | . (5.3) i=1 j=1 i=1 j=1 Taking the limit as i tends to infnity is equivalent to letting ci go to zero which

s log k results in 0 < µ(F ) ≤ H (F ) for s = − log c using (5.1), (5.2), and (5. 3). This gives s log k s that inf{s > 0 : H (F ) = 0} ≥ − log c . Since we know that as s increases, H (F ) log k gets smaller. Thus dimH (F ) ≥ s = − log c .

Theorem 5. Let F be an attractor for the IFS {S1,S2, ..., Sn} where |Si(x) −

Si(y)| = c|x − y|, where 0 < c < 1 and the open set condition holds. Then

log n dimH (F ) = dimS (F ) where s = − log c .

S n Proof. Since F is an attractor we know F = i=1 Si(F ). Further, since F is an attractor it is compact and thus closed. Since Si are contractions, they are also bijections and we have Si(F ) are closed. Thus F and Si(F ) are all Borel sets. Since

H s is a measure this gives

n n s s [ X s H (F ) ≤ H ( Si(F )) = H (Si(F )). (5.4) i=1 i=1

Pn s Pn s s Further, i=1 H (Si(F )) = i=1 c H (F ) since |Si(x) − Si(y)| = c|x − y|

Pn s by Theorem 4. Thus (5.4) becomes 1 ≤ i=1 c by dividing both sides of the equation by H s(F ) when H s(F ) > 0. This gives 1 ≤ ncs and by logarithms we

log n get s ≤ − log c where s is the Hausdor˙ Dimension of F . log n log n From the previous example we have − log c ≤ s. Thus s = − log c .

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