Fractal sets and dimensions

Examensarbete

Fractal sets and dimensions

Patrik Leifsson

Fractal sets and dimensions

Applied Mathematics, Linkopings Universitet Patrik Leifsson

LiTH - MAT - EX - - 06 / 06 - - SE

Examensarbete: 20 p Level: D

Supervisor: Jana Bjorn,

Applied Mathematics, Linkopings Universitet Examiner: Jana Bjorn,

Applied Mathematics, Linkopings Universitet Linkoping: May 2006

Matematiska Institutionen 581 83 LINK OPING SWEDEN May 2006 x x http://www.ep.liu.se/exjobb/mai/2006/tm/006/ LiTH - MAT - EX - - 06 / 06 - - SE

Fractal sets and dimensions

Patrik Leifsson

Fractal analysis is an important tool when we need to study geometrical objects less regular than ordinary ones, e.g. a set with a non-integer dimension value. It has developed intensively over the last 30 years which gives a hint to its young age as a branch within mathematics.

In this thesis we take a look at some basic measure theory needed to introduce certain denitions of fractal dimensions, which can be used to measure a set's fractal degree. Comparisons of these denitions are done and we investigate when they coincide. With these tools dierent fractals are studied and compared.

A key idea in this thesis has been to sum up dierent names and denitions referring to similar concepts.

box dimension, Cantor dust, Cantor set, dimension, fractal, Hausdor dimension, measure, Minkowski dimension, packing dimension, Sierpinski gasket, similarity, space-lling curve, topological dimension, von Koch curve.

Nyckelord Keyword Sammanfattning Abstract Forfattare Author Titel Title

URL for elektronisk version

Serietitel och serienummer Title of series, numbering

ISSN 0348-2960 ISRN ISBN Sprak Language Svenska/Swedish Engelska/English Rapporttyp Report category Licentiatavhandling Examensarbete C-uppsats D-uppsats  Ovrig rapport Avdelning, Institution Division, Department Datum Date

Abstract

Fractal analysis is an important tool when we need to study geometrical objects less regular than ordinary ones, e.g. a set with a non-integer dimension value. It has developed intensively over the last 30 years which gives a hint to its young age as a branch within mathematics.

In this thesis we take a look at some basic measure theory needed to intro-duce certain denitions of fractal dimensions, which can be used to measure a set's fractal degree. Comparisons of these denitions are done and we inves-tigate when they coincide. With these tools dierent fractals are studied and compared.

A key idea in this thesis has been to sum up dierent names and denitions referring to similar concepts.

Keywords: box dimension, Cantor dust, Cantor set, dimension, fractal, Haus-dor dimension, measure, Minkowski dimension, packing dimension, Sier-pinski gasket, similarity, space-lling curve, topological dimension, von Koch curve.

Acknowledgements

I would like to thank my supervisor and examiner Jana Bjorn for your constant support, guidance and patience. Your advice has been invaluable.

I would like to thank Nina for your support during this time. My opponent Daniel Petersson also deserves my thanks. Finally, I would like to thank my family and friends.

x

Preliminaries

In this section we collect some basic notations and denitions. x  y means that there exists c > 0 such that x

c < y  cx.

I denotes the set of irrational numbers. N denotes the set of natural numbers. Q denotes the set of rational numbers. R denotes the set of real numbers. Z denotes the set of integers.

Ball For e02 Rn and R 3 " > 0, we dene an open ball, B, as

B(e0; ") = fe 2 Rn : je e0j < "g;

and a closed ball, B, as

B(e0; ") = fe 2 Rn : je e0j  "g;

where e0 is the center and " the radius of the ball.

Open set, closed set A set A  Rn _{is open if there exists B(e; ")  A for all}

e 2 A. A set A is closed if Rn_{n A is open. A set A  E is open in E  R}n _if

for all x 2 A there exist a ball B(x; ") such that B(x; ") \ E  A \ E:

Bounded set The set E  Rn _{is bounded if the diameter of E, diam E, is}

bounded, i.e.

diam E = supfjx yj : x; y 2 Eg < 1:

Compact set A set E  Rn _{is compact if it is both closed and bounded.}

Closure of a set For a set E  Rn _{and a point x02 R}n _{we say that}

 x0 is an accumulation point of E if every ball B(x0; ") contains points

from E not equal to x0;

 r(E) is the set of all accumulation points of E;  the closure of E is dened as E = E [ r(E).

Topological base We say that a collection F of open sets in E  Rn _{is a}

topological base if for every open set G in E, there exists a subcollection G  F such that

G = [

F 2G

Contents

1 Introduction 1

1.1 Purpose of the thesis . . . 1

1.2 Structure of the thesis . . . 1

1.3 Dimensions . . . 2

2 The topological dimension 3 2.1 The small inductive dimension . . . 3

2.2 The large inductive dimension . . . 3

2.3 The covering dimension . . . 4

2.4 The topological dimension . . . 4

3 The Hausdor measure and dimension 7 3.1 The Hausdor measure . . . 7

3.2 The Hausdor dimension . . . 9

4 Minkowski dimensions 13 4.1 The packing dimension . . . 19

4.2 Product relations . . . 21

5 Fractals and self-similarity 23 5.1 Fractals . . . 23

5.2 Self-similarity . . . 23

6 Cantor sets 27 6.1 The ternary Cantor set . . . 27

6.2 Cantor set using ternary numbers . . . 28

7 The Sierpinski gasket 33 7.1 The Sierpinski gasket using the ternary tree . . . 36

7.2 The Sierpinski sieve . . . 36

8 The von Koch snow ake 39 8.1 The von Koch curve versus C2 _{. . . 42}

9 Space-lling curves 43 9.1 Peano space-lling curve . . . 43

9.2 The Heighway dragon . . . 45

10 Conclusions and nal remarks 49

Chapter 1

Introduction

A fractal can be described as an object less regular than "ordinary" geometrical objects. The term fractal came in use as late as 1975, by Mandelbrot who also gave one mathematical denition of what should be considered fractals. In this denition the use of fractal dimensions plays a big role and can be used to measure the fractal degree of a fractal, thereby allowing comparisons between dierent fractals. Though the denition is of relatively recent date, examples of sets now known as fractals in the sense of Mandelbrot date back to the late 19th century, e.g. the Weierstrass function

f(x) =X1

i=0

ai_cos(bi_{x); 0 < a < 1; ab > 1 +} 3

2;

which is continuous everywhere but nowhere dierentiable. Another classical example from this period is the triadic Cantor set, which will be studied thor-oughly later on in this thesis.

The use of fractal analysis is wide. It ranges from probability theory, phys-ical theory and applications, stock-market and to number theory among many others. Fractal objects and fenomena in nature such as mountains, coastlines and earthquakes is an area well studied by Mandelbrot. In the theory of fractal dimensions and fractals there is still much to be explored.

1.1 Purpose of the thesis

The purpose of this thesis is to sum up and investigate dierent theories and notations within some selected areas of fractal analysis in one comprehensible and well connected text. A reader with basic knowledge of abstract set theory and calculus should be able to enjoy most of the contents in this thesis.

1.2 Structure of the thesis

There are nine chapters (besides this introduction). In Section 1.3 conditions for dimensions that we will require to be fullled are dealt with. Chapters 2 - 4 deal with dierent types of dimensions and measures associated with them. In Chapter 4 we also extend our dimension algebra with some product relations

2 Chapter 1. Introduction

which are later on put to the test. In Chapter 5 we look at Mandelbrot's denition of a fractal set. The concepts of self-similarity and the similarity dimension are also studied here. These rst chapters cover the basic theory we need to study and compare dierent examples of fractals, beginning with a thorough study of the triadic Cantor set and its properties in Chapter 6. Two additional fractals are then introduced and studied in dierent perspectives in Chapters 7 - 8. In Chapter 9 we take a closer look at space-lling curves. Finally, all is rounded o with some conclusions and nal remarks in Chapter 10.

1.3 Dimensions

For a set A  Rn _{we will require the following to be satised, concerning the}

dimension dim A of A: (I)

8 < :

(i) dimfag = 0; where fag is the singleton set. (ii) dim I1_{= 1; where I}1 _{is the unit interval.}

(iii) dim Im_{= m; where I}m _{is the m-dimensional hypercube.}

(II) Monotonicity: If A  E then dim A  dim E.

(III) Countable stability: If fAig1i=1is a sequence of closed subsets of Rn, then

dim[1 i=1 Ai  = sup i1dim Ai:

(III') Finite stability: If A1; A2; : : : ; Amare closed subsets of Rn, then

dim m [ i=1 Ai  = max 1imdim Ai:

(IV) Invariance: If  : Rn _{! R}n _{is a homeomorphism, i.e. a continuous}

bijection whose inverse is continuous, then dim (A) = dim A. (IV') Lipschitz invariance: If g is a bi-Lipschitz transformation, i.e.

L1jx yj  jg(x) g(y)j  L2jx yj

for all x; y 2 A and some 0 < L1 L2< 1, then

dim g(A) = dim A:

Remark 1.1. A function g that fullls condition (IV') is also known as a lipeo-morphism.

Chapter 2

The topological dimension

There are three dierent denitions of the topological dimension: ind, Ind and Cov. The rst two are inductively dened. The covering dimension, Cov, is also known as the Lebesgue covering dimension or the topological dimension.

All of these dimensions coincide in a separable metric space and since we will restrict ourselves to subsets of Rn_{, we may consider any of the three dimensions}

as the topological dimension. We will denote the topological dimension of a set E by dimTE.

2.1 The small inductive dimension

Denition 2.1. The small inductive dimension of a set E  Rn _{is dened}

inductively as follows:

 ind ; = 1, where ; is the empty set.

 For an integer k  0, we have ind E  k if and only if there exists a topological base U for the open sets in E such that ind @U  k 1 for all U 2 U.

 We say that ind E = k if and only if ind E  k and ind E  k 1.  If ind E  k for all k  0, then ind E = 1.

2.2 The large inductive dimension

First we need the concept of separated sets.

Denition 2.2. For sets A; E  M we say that the set S  Rn _{separates A}

and E in M if there exist disjoint open sets V and W in Rn _{such that A  V ,}

E  W , and S = M n (V [ W ).

Denition 2.3. The large inductive dimension of a set E  Rn _{is dened}

inductively as follows:  Ind ; = 1.

4 Chapter 2. The topological dimension

 For an integer k  0, we have Ind E  k if and only if two disjoint closed sets in E can be separated in E by a set C such that Ind C  k 1.  We say that Ind E = k if and only if Ind E  k and Ind E  k 1.  If Ind E  k for all k  0, then Ind E = 1.

2.3 The covering dimension

As with the large inductive dimension we need to introduce a few terms before we dene the covering dimension.

Denition 2.4. A family F of subsets of Rn _{is a cover of a set E  R}n _if

E  S

F 2FF .

Denition 2.5. ([7] p. 95) If E and F are two covers of a metric space G and we have that for every F 2 F there is an E 2 E with F  E, then F is a renement of E.

Denition 2.6. For a family F of sets, the order, ord F, of F is less than or equal to k if and only if we have an empty intersection for any k + 2 of the sets. The order of F is equal to k if and only if ord F  k and ord F  k 1. Example 2.7. The family F = f(i 1; i + 1); i 2 Zg constitutes a cover of R by open intervals and ord F = 1.

Denition 2.8. The covering dimension of a set E  Rn _{is dened as follows:}

 Cov ; = 1.

 For an integer k  0, we have Cov E  k if and only if all nite open covers of E have an open renement with order less than or equal to k.  We say that Cov E = k if and only if Cov E  k and Cov E  k 1.  If Cov E  k for all k  0, then Cov E = 1.

2.4 The topological dimension

Denition 2.9. We say that

 E  Rn _{is dense in M  R}n _{if E = M.}

 M is separable if there exists a countable set E such that E = M. Proposition 2.10. Rn _{is separable.}

Proof. We rst show that if A = fa1; a2; : : : g and B = fb1; b2; : : : g are countable

then

A  B = f(a; b) : a 2 A; b 2 Bg

is countable. This is easily proved e.g. by the Cantor diagonalization method. Simply arrange the numbers ai in a horizontal list versus the numbers bi in a

2.4. The topological dimension 5

numbers (ai; bj), i.e. from (a1; b1) we move to (a2; b1), then to (a1; b2), from

this point we move to (a1; b3) and then to (a2; b2) and so on. In this manner

we can count all the numbers (ai; bj) in the list and thus conclude that A  B

above is countable.

Now, since Z is countable and there is an injection from Q to Z  Z, it follows that Q is countable (in accordance with the Schroder-Bernstein theorem [3] p. 100), Q2_{= Q  Q is countable, and by induction, Q}n _{is countable for all}

n. Thus Rn _{is separable since R}n_{= Q}n _{for all n.}

Now, what is interesting for us is that for a separable set ind, Ind and Cov coincide. Since we restrict ourselves to subsets of Rn_{matters are now simplied}

a bit.

Theorem 2.11. (Theorem 8.10 in [1]) For every separable set E, ind E = Ind E = Cov E:

Corollary 2.12. If E  Rn_{, then}

ind E = Ind E = Cov E:

Thus for a subset E of Rn_{, we need only think of the topological dimension}

as one dimension, and we dene the topological dimension of E as dimTE = ind E:

Denition 2.13. The set E is said to be totally disconnected if for any e1; e22 E, e16= e2, we have that e1 and e2 can be separated by the empty set.

Proposition 2.14. ([6], Theorem A.4.13) A compact set E  Rn _has

dimTE = 0 if and only if E is totally disconnected.

Proposition 2.15. The topological dimension, dimT, fullls the dimension

re-quirements stated in Section 1.3.

Proof. (I)(i) For the singleton set fag we have dimTfag = 0 by Proposition 2.14.

(I)(ii) The open intervals

Ix;"= (x "; x + "); x 2 I1= I; " > 0;

constitute a topological base for the interval I. Now since dimT@Ix;"= dimTfx "; x + "g = 0

by (i), we have that dimTI  1 and the fact that I is connected gives us

dimTI 6= 0 and thus dimTI = 1.

(I)(iii) That dimTIm= m can be shown by an argument similar to that for

(I)(ii).

(II) Let A  E  Rn _{and dim}

TE = k. Thus we have a topological base U

of open sets for E with dimT@U  k 1 for all U 2 U. Thus the collection

fU \ A : U 2 Ug forms a topological base for A. Since

6 Chapter 2. The topological dimension

we obtain dimTA  k = dimTE.

(III) See e.g. Theorem 3.7 in [1].

(IV) (See e.g. Theorem 3.1.6 in [7]). This can intuitively be understood by studying a topological base U for E. Since f and f 1_{are continuous, it follows}

that

U0_{= ff(U) : U 2 Ug}

is a topological base for f(E). Shortly, the base is preserved under the mapping f and the inversion f 1_{. From this it can be shown that}

dimTf(E)  dimTE;

and similarly for the opposite inequality.

Remark 2.16. In condition (III) the fact that the Ai's are closed sets is

impor-tant. For sets Ai that are not closed, the condition is not true in general.

Let us illustrate Remark 2.16 with an example.

Example 2.17. Let I and Q be denoted as earlier. For q1and q2both in Q such

that q1< q2, the empty set separates them in Q. Thus Q is totally disconnected

by Denition 2.13 and hence

dimTQ = 0

by Proposition 2.14. In an analogous manner with the same references it follows that

dimTI = 0:

So we have that

dimTI = dimTQ = 0;

but

dimT(I [ Q) = dimTR = 1 6= maxfdimTQ; dimTIg:

Proposition 2.18. (Theorem 3.2.10 in [7]) If A; E  Rn_{, then}

dimT(A [ E)  1 + dimTA + dimTE:

Example 2.17 shows that this connection cannot be improved. Proposition 2.19. (Theorem A.4.14 in [6]) dimTRn= n.

Remark 2.20. In this chapter we have seen examples of dierent methods to calculate a set's topological dimension. The key idea to calculate dimTE for a

set E is to study @E as we have seen, e.g. a line I has topological dimension 1 since @I consist of the endpoints of the line which have topological dimensions 0. A square can be enclosed by a closed curve with topological dimension 1 and thus the square has topological dimension 2 and so on.

Chapter 3

The Hausdor measure and

dimension

3.1 The Hausdor measure

Denition 3.1. We say that fEig1i=1 is a -cover of a set E if E  1

S

i=1Ei and

0  diam Ei , for all i. For a set E  Rn, s  0, and  > 0, dene

Hs (E) = inf 1 X i=1 (diam Ei)s;

where the inmum are taken over all (countable) -covers fEig1i=1 of E. Also

dene the s-dimensional Hausdor measure by Hs_{(E) = lim} !0H s (E) = sup >0H s (E):

To put it in words, the Hausdor measure approximates a sets lenght, area or volume through covers with diameters less than or equal to . The letter s denotes what is approximated, i.e. lenght, area or volume. The approximation gets better the smaller sets we use in the covering which makes it natural to let  ! 0 in the denition.

Denition 3.2. An outer measure, , on Rn _{is a positive set function on all}

subsets of Rn _{that satises:}

 (;) = 0;

 monotonicity: (A)  (E) if A  E  Rn_;

 countable sub-additivity:  1 [ i=1 Ei   1 X i=1 (Ei) for all Ei  Rn. Proposition 3.3. Hs  is an outer measure. Leifsson, 2006. 7

8 Chapter 3. The Hausdor measure and dimension

Proof. With the above denition of outer measure we have  Hs

(;) = 0 since diam ; = 0 .

 Let A  E  Rn _{and " > 0. Then there is a -cover fE}

ig11=1 of E such

that ₁

X

i=1

(diam Ei)s Hs(E) + ":

But fEig11=1 is also a valid -cover of A and thus

Hs (A)  1 X i=1 (diam Ei)s Hs(E) + ":

Now, letting " ! 0 gives us Hs

(A)  Hs(E):

 Assuming P1

i=1H s

(Ei) < 1 we have for an arbitrary " > 0 that for each

i  1 there exists a -cover fAi

jg1j=1 of Ei such that 1 X j=1 (diam Ai j)s< Hs(Ei) + 2 i": Thus fAi

jgi;j1constitutes a valid -cover of 1 S i=1Ei and X i;j1 (diam Ai j)s 1 X i=1  Hs (Ei) + 2 i"  =X1 i=1 Hs (Ei) + ":

Letting " ! 0 proves the claim.

Proposition 3.4. Hs_{is a measure, i.e. if fA}

ig1i=1 is a pairwise disjoint

count-able collection of measurcount-able sets, then  1 [ i=1 Ai  = 1 X i=1 (Ai): (3.1)

Proof. See for example Theorem 4.2 in [16]. Remark 3.5. Hs

 is an outer measure though not a measure.

Let us illustrate the truth of Remark 3.5.

Example 3.6. We will consider the set E = (Q \ [0; 1])  [0; 1]. If we let Ai= fqig  [0; 1], for each qi in Q \ [0; 1], we have

H1 (Ai) = inf 1 X j=1 (diam E_ij)1_{= 1; (3.2)}

3.2. The Hausdor dimension 9

where E_ij are -covers of Aiwith diam Eij . Since we are considering

Carte-sian products fqig  [0; 1] we have 1

X

j=1

(diam E_ij)  1

and thus (3.2) is fullled. Then using balls, Bi, of radii  to cover E we need

approximately c
1

2 of these, where c 2 R. Thus

H1  _[1 i=1 Ai  = infX1 i=1 (diam Bi)  c
_1₂
  1_ < 1 X i=1 H1 (Ai) = 1:

Hence the necessary condition (3.1) is not fullled in order for Hs

 to be a

measure.

Denition 3.7. The Lebesgue measure Ln _{on R}n _{is dened as follows. For A}

of the form

A = f(x1; : : : ; xn) 2 Rn: ai xi big (3.3)

dene

Ln_{(A) = (b}

1 a1)(b2 a2)


(bn an);

and extend Ln _{to general subsets of R}n _by

Ln_{(A) = inf}nX1 i=1 Ln_(A i) : A  1 [ i=1 Ai o

for all Ai of the form (3:3):

Remark 3.8. It can be proved that when s = n on Rn_{, then H}n _{is the Lebesgue}

measure (within a constant multiple). (See for example Theorem 30 in [19]).

3.2 The Hausdor dimension

Turning our attention once again towards dimensions, we can now with the aid of the s-dimensional Hausdor measure dene the Hausdor dimension. Denition 3.9. The Hausdor dimension of a set E  Rn _is

dimH(A) = supfs : Hs(E) > 0g = supfs : Hs(E) = 1g

= infft : Ht_{(E) < 1g = infft : H}t_{(E) = 0g:}

Remark 3.10. The previous denition can also be expressed as Hs_{(E) =} 1 if s< dimH(E);

0 if s> dimH(E);

and Hs_{(E) can attain any value in [0; 1] for s = dim} H(E).

In other words, dimHE is the critical value where the s-dimensional

Haus-dor measure of the set E so to speak jumps from innity to zero.

Proposition 3.11. The Hausdor dimension, dimH, satises the conditions

10 Chapter 3. The Hausdor measure and dimension

Proof. We again treat each condition separately.

(I)(i) We have dimHfag = 0 for the singleton set fag, since diamfag = 0 gives us

Hs_{(fag)  (diamfag)}s_{= 0}

for all s > 0, and thus

dimHfag = inffs : Hs(fag) = 0g = 0:

(I)(ii) We have 0 < H1_(I1_{) < 1 and hence dimHI}1_{= 1, using Remark 3.10.}

(I)(iii) Let Im_{be an m-dimensional hypercube in R}n_{, 1  m  n. We have}

Hm_(Im_{) = cL}m_(Im_{) 2 (0; 1):}

Hence dimHIm= m, using Remark 3.10.

(II) Monotonicity: Let A  E  Rn_{. Since H}s_{(A)  H}s_{(E) when}

A  E  Rn_{, it follows from Denition 3.9 that dimH(A)  dimH(E).}

(III) Countable stability: From the monotonicity we have for all j, dimH _[1 i=1 Ei   dimHEj

and hence, taking supremum over all j, dimH _[1 j=1 Ej   sup j1dimHEj: The inequality dimH _[1 i=1 Ei   sup j1dimHEj

follows from the fact that if s > dimHEi for all i, then Hs(Ei) = 0

for all i, which gives us Hs1S i=1Ei  = 0 and hence dimH _[1 i=1 Ei   s:

(IV') Lipschitz invariance: See for example Chapter 2 in [9].

Remark 3.12. It can be shown that the von Koch curve (see Chapter 8) which has Hausdor dimension log 4

log 3is homeomorphic to [0,1] with Hausdor dimension

1 and thus we can directly see that invariance is not fullled for the Hausdor dimension, (see e.g. [17]).

3.2. The Hausdor dimension 11

Proof. This is justied by the fact that for a singleton set Ei we have

H0_(E i) = 1:

Thus

dimH(Ei) = 0:

Hence, by the countable stability dimH

[1 i=1Ei

 = 0:

Chapter 4

Minkowski dimensions

Other commonly used names for the Minkowski dimension are e.g. box-counting dimension, box dimension, fractal dimension, metric dimension, capacity dimen-sion, entropy dimendimen-sion, logarithmic density and information dimension. The logic in these names can often be seen through their context. We will however in favor of simplicity only use Minkowski dimension to mean any of these.

The Minkowski dimension of a non-empty bounded subset of Rn _{is dened}

through an upper and lower dimension, which need not coincide.

Denition 4.1. For a non-empty bounded subset E of Rn _{we dene the upper}

Minkowski dimension as

dimME = inffs : lim sup

"!0+ N(E; ")" s_{= 0g;}

where 0 < " < 1 and N(E; ") is the least number of balls with radius " needed to cover E. In a similar manner we dene the lower Minkowski dimension as

dim_ME = inffs : lim inf

"!0+ N(E; ")" s_{= 0g:}

Instead of using the covering numbers N(E; ") one can also use the packing numbers

P (E; ") = maxfk : 9 disjoint balls B(xi; "); i = 1; : : : ; k; with xi2 Eg: (4.1)

Proposition 4.2. For all E  Rn _{the following holds}

N(E; 2")  P (E; ")  N(E; "=2): Proof. To convince ourselves of the validity of

N(E; 2")  P (E; ");

let k = P (E; ") and consider the disjoint balls B(xi; "), xi2 E, i = 1; : : : ; k.

Now, if there exists an x in E nSk_i=1B(xi; 2") then the balls

B(x1; "); : : : ; B(xk; "); B(x; ") are pairwise disjoint and thus

k + 1  P (E; ") = k

14 Chapter 4. Minkowski dimensions

which is a contradiction. So the balls B(xi; 2") cover E and thereby

N(E; 2")  k = P (E; "): For the validity of

P (E; ")  N(E; "=2);

let k1= N(E; "=2) and k2= P (E; ") and let x1; : : : ; xk1 2 Rn and

y1; : : : ; yk2 2 E be such that

E [k1

i=1

B(xi; "=2)

and the balls B(yl; "), l = 1; : : : ; k2, are disjoint. This results in that all of the

yl's are in some B(xi; "=2) and no B(xi; "=2) have more than one point yl(since

the balls B(yl; ") are disjoint). This gives us k2 k1 and thus

P (E; ")  N(E; "=2):

It follows from Denition 4.1 and Proposition 4.2 that dimME = lim sup

"!0+

log N(E; ")

log(1=") = lim sup"!0+

log P (E; ")

log(1=") (4.2) and

dim_ME = lim inf

"!0+

log N(E; ")

log(1=") = lim inf"!0+

log P (E; ")

log(1=") : (4.3) Let us show the rst equality in (4.2). The second equality in (4.2) follows from Proposition 4.2. Suppose s > dimME. Then for all  there exists an "0 > 0

such that

N(E; ")"s_{< ; for all " < "} 0:

Then

log N(E; ")"s_{< log }

so

s >log  log N(E; ")_{log "} ;

and letting " ! 0+ and taking inmum over all s > dimME, we get

dimME  lim sup "!0+ log N(E; ") log(1=") : Now suppose lim sup "!0+ log N(E; ") log(1=") < s < dimME: Then there exist "j! 0 and 0> 0 such that

15

which implies

log N(E; "j) + s log "j> log 0:

But then

s  lim sup

"j!0+

log N(E; "j) log 0

log(1="j)  lim sup"!0+

log N(E; ") log(1=") ; which is a contradiction. Thus

dimME = lim sup "!0+

log N(E; ") log(1=") :

When (4.2) and (4.3) are equal, the common value is called the Minkowski dimension of E, and we write

dimME = lim_"!0log N(E; ")_log(1=") = lim_"!0log P (E; ")_log(1=") : (4.4)

In this denition of the Minkowski dimension, the number N(E; ") can be re-placed by any of the following numbers:

 the smallest number of closed balls of radius " needed to cover E;  the smallest number of cubes of side " needed to cover E;

 the number of "-mesh cubes that intersect E, (where an "-mesh cube is a cube of the form [e1"; (e1+ 1)"] 


 [en"; (en+ 1)"], e1; : : : ; en are

integers);

 the smallest number of sets with diameter at most " covering E.

Arguments similar to the proof of Proposition 4.2 show that this leads to the same denotation.

The Minkowski dimension can also be dened by means of the n-dimensional volume of an "-neighbourhood of E  Rn_{. The "-neighbourhood, E}

", of E is

dened as

E"= fx 2 Rn: jx yj  " for some y 2 Eg: (4.5)

Then, using the above dened Lebesgue measure, we have the following equiv-alent formulas for the Minkowski dimension.

Proposition 4.3. Let E  Rn_{. Then}

dimME = n + lim sup "!0+

log Ln_(E ")

log(1=") ; (4.6)

dim_ME = n + lim inf

"!0+

log Ln_(E ")

log(1=") ; (4.7)

and

dimME = n + lim_"!0+log L n_(E

")

log(1=") (4.8)

16 Chapter 4. Minkowski dimensions

Proof. First we prove that the inequalities c"n_{P (E; ")  L}n_(E

")  c(2")nN(E; ") (4.9)

hold, where c is the volume of the unit ball in Rn_{. If we have a cover of E by}

N(E; ") balls with radii ", then we have that E" can be covered by balls with

radii 2". Thus Ln_(E ")  c(2")nN(E; "): To understand that c"n_{P (E; ")  L}n_(E ");

simply notice that the space that the P (E; ") disjoint balls ll is covered by the "-neighbourhood of E and the n-dimensional Lebesgue measure of E" exceeds

or equals c"n_{P (E; ") thereby.}

Now, using (4.9), we have dimME = lim sup

"!0+

log P (E; ")

log(1=")  lim sup_"!0+

log(c 1_" n_Ln_(E ")) log(1=") = lim sup "!0+  log Ln_(E ") log c n log " log "  = lim sup "!0+  log Ln_(E ") log " + n  = n lim inf "!0+ log Ln_(E ") log " : This was attained from c"n_{P (E; ")  L}n_(E

"). Using the last inequality in (4.9)

we have

dimME = lim sup "!0+

log N(E; ")

log(1=")  lim sup"!0+

log(c 1_(2") n_Ln_(E ")) log(1=") = lim sup "!0+ 

log Ln_{(E") log c n log(2")}

log "  = lim sup "!0+  log Ln_(E ") log " + n  = n lim inf "!0+ log Ln_(E ") log " : Thus we conclude that

dimME = n lim inf "!0+

log Ln_(E ")

log " ;

which proves (4.6). Now, (4.7) follows in an manner analogous to the above. Finally, dimME = n + lim "!0+ log Ln_(E ") log(1=") follows from (4.6) and (4.7).

17

From the denition of the Hausdor measure we can deduce the useful re-lation Hs

(E)  N(E; )s. In other words, if fAig1i=1 is a -cover of E, then as

diam Ai  for all i, we have 1 X i=1 (diam Ai)s _|s+ s+


+ _{z s_} N(E;) = N(E; )s_:

Theorem 4.4. For all E  Rn_,

dimHE  dimME  dimME: (4.10)

Proof. If s < dimHE, then

0 < Hs_{(E) = lim} !0+H

s

(E)  lim_!0+N(E; )s

and thus log N(E; ) + s log  > log Hs_{(E) 1, for small enough  > 0. From}

this we have

s  lim inf

!0+

log N(E; ) log Hs_(E)

log(1=) = dimME  dimME:

Taking supremum over all s gives us the desired inequality.

That the inequalities in Theorem 4.4 may be strict is best shown with an example.

Example 4.5. A (compact) set needs not have its Hausdor dimension equal to its Minkowski dimension. The set E = f0; 1;1

2;13;14; : : :g has Hausdor

di-mension 0 since it is countable. However, dimME = 1₂. To show this, let " > 0

and k be the smallest integer such that 1 k 1 1 k = 1 k(k 1) < ": A rst-order approximation of " in terms of k is "  1

k2. The number of balls of

radius " it takes to cover the points 1;1

2;13;14; : : : ;k 11 equals k 1  p1". And

to cover the points which lie in E \ [0;1

k] by balls of radius ", it takes about 1

2k" 2p1" balls. Hence the number of balls needed to cover E is essentially

N(E; ") ₂p1 "+ 1 p "  1 p " from which we obtain

dimME = _"!0+lim log N(E; ")_{log "} = _"!0+lim

log(_p1 ") log " = lim"!0+ 1 2log " log " = 1 2: A question that still remains is whether the Minkowski dimension fullls the dimension requirements from Section 1.3.

Proposition 4.6. The upper Minkowski dimension, dimM, satises conditions

(I), (II), (III') and (IV') from Section 1.3. The lower Minkowski dimension, dim_ME, fullls conditions (I), (II) and (IV') of these.

18 Chapter 4. Minkowski dimensions

Proof. (I)(i) We have

dimMfag = inffs : lim sup

"!0+ N(fag; ")" s_{= 0g:}

Since N(fag; ") = 1 for all " > 0, we have for all s > 0 that lim sup

"!0+ N(fag; ")" s_{= 0}

and hence 0  dim_Mfag  dimMfag = 0.

(I)(ii)-(iii)We have N(Im_{; ")  "} m_{and hence}

dim_MIm_{= dim}

MIm= dimMIm= lim_"!0+log N(I m_{; ")}

log(1=") = m:

(II) If A  E then the number of balls with radius " needed to cover A at most equals the number of balls with radius " needed to cover E, i.e.

N(A; ")  N(E; "): Thus we have

dim_MA  dim_ME and dimMA  dimME

and so dimMA  dimME by (4.2){ (4.4).

(III') Let fEijg1j=1be a -cover of Ai for i = 1; : : : ; n. Then

fEij; i = 1; : : : ; n; j = 1; : : : ; 1g is a -cover ofSn_i=1Ai, so N[n i=1 Ai;   Xn i=1 N(Ai; ):

If s > dimMAi for all i, then we have in accordance with Denition 4.1 that

lim sup

!0+ N(Ai; ) s_{= 0:}

Adding over i = 1; : : : ; n gives us lim sup !0+ N _[n i=1 Ai;   s_{= 0:}

Thus from Denition 4.1 we have dimM(Sni=1Ai)  s. Hence

dimM( n

[

i=1

Ai)  max_i=1;:::;ndimMAi:

Now, suppose that dimM(Sni=1Ai) < max_i=1;:::;ndimMAi, i.e. there is an i such

that dimMAi> dimM _[n i=1 Ai  :

4.1. The packing dimension 19

But this contradicts the monotonicity. Thus dimM _[n i=1 Ai  = max i=1;:::;ndimMAi:

(IV') The Lipschitz invariance is fullled due to the fact that if jg(x) g(y)j  Ljx yj

and the set E can be covered by N(E; ") sets with diameter less than or equal to ", then the images of these N(E; ") sets form a cover of g(E) by sets with the diameter less than or equal to L", so that N(g(E); L")  N(E; ").

This shows that

dimMg(E)  dimME and dimMg(E)  dimME;

using (4.2) and (4.3). Applying this same argument to g 1_{instead, gives us the}

opposite inequality, ([9], p. 44).

Example 4.7. Let us now consider yet another set and calculate its Hausdor and Minkowski dimensions. The set to consider is F = f0; 1;1

2;14;18; : : :g. Since

F  E, where E is the set considered in Example 4.5, we immediately know that dimMF  dimME = 1₂ from the monotonicity condition. We also know

that dimHF = 0 since F is countable as well. Now, with 0 <  = 2 n we we

get a valid -cover of F by N(F; 2 n_{) = n + 2 balls. Since}

lim n!1 log N(F; 2 n₎ log 2n  lim_n!1 log n log 2n = lim_n!1 log n n log 2= 0; we have that dimMF = dimMF = 0:

Now consider the mapping

 : x !1₂1x 1; from E to F for each nonzero element x in E. Since

lim x!0 1 2 1 x 1 = 0; we have (0) = 0 and thus the mapping is continuous.

Remark 4.8. Examples 4.5 and 4.7 shows that the countable stability (III) and the invariance (IV) fail for the Minkowski dimension.

4.1 The packing dimension

As we have seen before (Example 4.5), the Minkowski dimension does not satisfy the countable stability criterion for dimensions. Neither is the nite stability criterion fullled for dim_M. However, these complications can be overcome if we introduce the packing dimension for a set E  Rn_.

20 Chapter 4. Minkowski dimensions

Denition 4.9. The lower and upper packing dimensions are dim_PE = infnsup

i1dimMEi

o

and dimPE = inf

n sup

i1dimMEi

o ; where the inma are taken over all countable covers fEig1i=1 of E  Rn by

bounded sets.

With this denition, we get dim_PE = dimPE = 0 when E is countable, and

the countable stability criterion for the packing dimension is fullled. The other dimension requirements valid for the Minkowski dimension are still valid. Denition 4.10. Let 0 < " < 1 and 0  s < 1. For a set E  Rn _{we dene}

Ps

"(E) = sup 1

X

i=1

(diam Bi)s and Ps(E) = lim_"!0+P"s(E) = inf_">0P"s(E):

The supremum is taken over all collections of disjoint balls fBig of radius less or

equal to " and centers in E. We then dene the s-dimensional packing measure of a set E  Rn _as Ps_{(E) = inf}nX1 i=1 Ps_(E i) : E = 1 [ i=1 Ei o : Proposition 4.11. Ps_{(E) is a measure on R}n_.

Proof. See for example [16] p. 82.

We now use the s-dimensional packing measure to dene the packing dimen-sion.

Theorem 4.12. For a set E  Rn_,

dimPE = inffs : Ps(E) = 0g = inffs : Ps(E) < 1g

= supfs : Ps_{(E) > 0g = supfs : P}s_{(E) = 1g:}

Proof. See Theorem 5.11 in [16].

Denition 4.13. For E  Rn_{, we dene the packing dimension as}

dimPE = dimPE:

Proposition 4.14. If E  Rn _then

dimTE  dimHE  dimPE  dimME:

Proof. The rst inequality follows from Proposition 5.1. The second inequality, i.e. dimHE  dimPE, follows from the fact that

Hs_{(E)  P}s_{(E) for all E  R}n_;

(see Theorem 5.12 in [16]), and thus we have dimHE  dimPE

4.2. Product relations 21

due to Denition 3.9 and Theorem 4.12. The following proof of the last inequal-ity can be found on p. 46 in [9].

For arbitrary t and s such that t < s < dimPE we have that

Ps_{(E)  P}s_{(E) = 1:}

So for 0 <   1 there are disjoint balls fBig1i=1 with centers in E and radii at

most equal to  such that

1 <

1

X

i=1

(diam Bi)s:

Next we assume that for all k we have that nk of these balls satisfy

2 k 1_{< diam B} i 2 k: Then 1 < 1 X k=0 nk2 ks (4.11)

is also satised. Now, unless we want (4.11) contradicted there need to be some k with

nk > 2kt(1 2t s)

as we also sum over k. The nk balls all have centers in E and we can shrink

them to have radii 2 k 1_{< .}

Hence P (E; 2 k 1_{)  n} k and (2 k 1₎t
_{P (E; 2} k 1_{)  n} k(2 k 1)t> 2 t(1 2t s); where 2 k 1_{< . Thus} lim sup !0 P (E; ) t_{ 2} t_{(1 2}t s_{) > 0}

so dimME  t, for all t < dimPE and the claim follows thereby.

The following proposition gives a sucient condition for when the packing and Minkowski dimensions coincide.

Proposition 4.15. (Corollary 3.9 in [9]) If E  Rn _{is a compact set such that}

dimM(E \ G) = dimME;

for all open sets G intersecting E, then

dimPE = dimME:

4.2 Product relations

There are some valuable formulas which can reduce the amount of eort needed to calculate the dimension. We shall now consider some of them.

22 Chapter 4. Minkowski dimensions

Proposition 4.16. For sets A; E  Rn _{we have}

dimH(A  E)  dimHA + dimHE; (4.12)

dimH(A  E)  dimHA + dimME; (4.13)

dimM(A  E)  dimMA + dimME; (4.14)

Proof. See Chapter 7.1 in [9].

There is also a product formula concerning the topological dimension of sets. Proposition 4.17. (Theorem 3.9 in [1]) Let A  Rn _{and E  R}n _{be two sets,}

not both empty. Then

Chapter 5

Fractals and self-similarity

5.1 Fractals

Proposition 5.1. ([12], p. 3) For any set E we have dimTE  dimHE.

A complete proof will not be given. Some things can be noted however. For E = ; we obviously have dimTE < dimHE since dimT; = 1 and dimHE  0.

Remembering Proposition 3.13 we know that for a countable set E we have dimHE = 0. Referring to Proposition 2.14, which says that a compact totally

disconnected set E  Rn _{has dim}

TE = 0, we have these cases covered too. The

cases left to study we leave unproven. (See e.g. p. 104 in [13]). Denition 5.2. We say that a set E  Rn _{is fractal if dim}

TE < dimHE. The

fractal degree of the set E is (E) = dimHE dimTE.

Proposition 5.3. A set E is fractal if the value of dimH(E) is non-integer.

Proof. The proposition follows from the fact that dimT(E) only takes on integer

values, so if dimH(E) is not integer then neither is (E).

5.2 Self-similarity

A self-similar set is loosely speaking a set consisting of scaled copies of itself. Denition 5.4. For a closed set E  Rn_{, the mapping T : E ! E is called a}

contraction on E if there is a c 2 (0; 1) such that

jT (x) T (y)j  cjx yj (5.1)

for all x; y 2 E.

The smallest c satisfying (5.1) is called the contraction ratio of T . Moreover, a contraction is a continuous mapping.

Denition 5.5. A xed point of a mapping T : E ! E is a point x 2 E that remains unchanged under the mapping, i.e. T x = x.

The following proposition is proved in [15] p. 323.

24 Chapter 5. Fractals and self-similarity

Proposition 5.6. Let E 6= ; be a closed set with the contraction T : E ! E dened on it. Then T has precisely one xed point.

When we have equality in (5.1), then T preserves the geometrical similarity, and we call T a similarity or simlitude. For the smallest c fullling (5.1) we call T a similar contraction.

Denition 5.7. For a family T = fT1; T2; : : : ; Tmg of similarities with

con-traction ratios c1; c2; : : : ; cm, m  2, we say that a nonempty compact set E is

invariant under T if E = m [ i=1 Ti(E):

Proposition 5.8. For any T as in Denition 5.7 there is a unique invariant set.

Proof. See [12] p. 19.

An invariant set under a family of similarities T is called a self-similar set. Denition 5.9. We say that the contractions T1; T2; : : : ; Tm fulll the open set

condition if there is a nonempty bounded open set O such that

m

[

i=1

Ti(O)  O

with the Ti(O)'s pairwise disjoint.

With the prerequisites thus far gained, we introduce yet another dimension concept.

Denition 5.10. Let E be a self-similar set such that E = T1(E) [ T2(E) [


[ Tm(E);

where Ti, i = 1; : : : ; m, are similarities with contraction ratios ci 2 (0; 1), and

the Ti(E)'s are disjoint. The similarity dimension of E, dimSE, is the unique

solution s to the Moran equation cs

1+ cs2+


+ csm= 1: (5.2)

In the special case when

c1= c2=


= cm= c

we have that

mcs_{= 1 and hence log m + s log c = 0:}

Thus

dimSE = s = _log(1=c)log m : (5.3)

Despite its name, the similarity dimension does not satisfy the dimension conditions from Section 1.3 in general. However, under certain circumstances it coincides with the other dimensions, thus justifying its notion as a dimension.

5.2. Self-similarity 25

Proposition 5.11. (Theorem 2.7 in [10], Theorem 4.14 in [16]) Let Ti be

sim-ilarities on Rn _{satisfying the open set condition with contraction ratios ci,}

i = 1; 2; : : : ; m. If E is the invariant set of fTigmi=1, then

 dimHE = dimPE = dimME = dimSE;

 0 < Hs_{(E) < 1 and P}s_{(E) < 1, where s = dimSE;}

 There exist e1; e22 (0; 1) such that for s = dimSE,

e1rs Hs(E \ B(x; r))  e2rs

for all x 2 E and 0 < r  1.

Remark 5.12. If the open set condition is not fullled in Proposition 5.11, then we instead get the relation

dimHE = dimPE = dimME  dimSE:

(See e.g. [12], Theorem 2.3).

Proposition 5.13. (Proposition 9.6 in [9]). For contractions T1; T2; : : : ; Tm

with contraction ratios ci < 1; i = 1; : : : ; m, on a closed invariant set E  Rn

we have that

dimHE  s and dimME  s;

Chapter 6

Cantor sets

Generally, for 0 <  < 1

2, we dene Cantor sets on R as the limit set

C() = 1 \ i=0 2i [ j=1 Ei;j

where E0;1 = [0; 1]; E1;1 = [0; ]; E1;2 = [1 ; 1], and for dened intervals

Ei 1;1; : : : ; Ei 1;2i 1, the intervals E_i;1; : : : ; E_i;2i are dened through removing

intervals of length (1 2) diam Ei 1;j = (1 2)i 1from the middle of each

interval Ei 1;j. Thus each Ei;j has length i.

The following proposition will be veried in the following sections. Proposition 6.1. Some important properties of C() are the following:

 It is uncountable, compact and totally disconnected.  L(C()) = 0.

 dimHC() = _log(1=)log 2 .

 Hs_{(C()) = 1, where s = dim}

HC().

6.1 The ternary Cantor set

Among the dierent choices of  for C(),  = 1=3 is the most frequently used one. We shall therefore show some of the general properties of Cantor sets for this one to make it less abstract. The general case can be treated in an analogous manner. The set

C(1 3) = 1 \ i=0 2i [ j=1 Ei;j; where E0;1 = [0; 1]; E1;1= [0;1₃]; E1;2= [2₃; 1]

and so on is called the ternary or triadic Cantor set or simply the Cantor dust. Following the procedure recently described we start with the unit interval. If we denote C(1=3) with just C, we have that the unit interval is our set C0. To

receive C1we remove the open middle third interval from C0, i.e.

C1= C0n (1₃;2₃) = [0;1₃] [ [2₃; 1]:

28 Chapter 6. Cantor sets

From each of these two intervals we then remove the open middle third intervals

of length _

1 2
1₃1₃2 1= 1₉: Thus

C2= C1n ((1₉;2₉) [ (7₉;8₉)) = [0;1₉] [ [2₉;1₃] [ [2₃;7₉] [ [8₉; 1]:

Continuing like this in innitely many steps gives us our limit set C =

1

\

i=0

Ci:

This set is obviously quite porous containing no intervals of positive length, hence its name Cantor dust. A simple illustration of the rst dierent genera-tions Ci of C follows below.

Figure 6.1: The rst generations of the triadic Cantor set.

6.2 Cantor set using ternary numbers

An alternative view of the triadic Cantor set is with base three, i.e. ternary, expansions of each number in [0; 1]. First of all, any number can be written with a dierent expansion than the one it already has. Consider for example the base two expansion of the number seven (written in base ten), i.e. 7 = 22_{+ 2}1_{+ 2}0

so we have 710= 1112. Another example is 810= 23= 10002. In an analogous

manner we can rewrite any fraction written in base n into another base m 6= n. E.g. 45 64 = 1 2 + 1 8+ 1 16+ 1 64 = 0:1011012:

What we need to know in our further investigation is how base three expansion works. This is accomplished in a similar fashion as converting base ten into base two. For instance,

710= 2
31+ 30= 213 and 810= 2
31+ 2
30= 223:

Likewise we convert the base ten fraction 4=7 as 0:120102120 : : :3= 0:1201023,

where the underline indicates a repeating decimal expansion of the underlined digits. Now, looking at the Cantor dust from a base three expansion point of view gives us what is illustrated below.

Thus, the ternary Cantor set consists of all numbers between zero and one with ternary expansions in zeros and twos only. Some numbers have two dierent expansions, e.g.

1

6.2. Cantor set using ternary numbers 29

Figure 6.2: Base three expansion of the triadic Cantor set.

but in these cases it is most important whether the number can be written with only zeros and twos, because then it counts as a member of the set even if it also has its expansions using zeros and ones. The points 0 = 0:0₃ and 1 = 0:2₃ evidently count as well. Let us verify some of C's properties mentioned earlier. Proposition 6.2. The Cantor set C is uncountable.

Proof. Consider the second generation C1of C and the binary sequences (xi)1i=1

with xi 2 f0; 1g. Now, for every c 2 C we let x1 = 0 if c belongs to the left

segment of C1 and x1 = 1 if c is found in the right segment of C1. After this

step is done we now need to consider in which of the two possible segments of C2's four parts c is in. Letting this procedure continue further yields a

binary sequence (x1; x2; : : :) for each c 2 C. Similarly each of those sequences

corresponds to a c in C. Thus we have a bijection between C and the binary sequences (xi)1i=1. Since the set of binary sequences is uncountable, so is C.

Proposition 6.3. The (triadic) Cantor set has dimT = 0.

Proof. The Cantor set is compact since it is both closed and bounded. It is closed since in C =T1_i=0Cneach Cnconsists of a nite union of closed intervals

and using the fact that the union of a nite collection of closed sets is closed according to De Morgan's laws and that any intersection of closed sets is closed. To see that C is totally disconnected, assume that c1; c22 C and c1 < c2. Let

 = c2 c1. Each interval Cn 2 C is of length 3 n. Choosing n such that

3 n _{<  places c1 and c2 in dierent intervals. Supposing I = [a; b] is the last}

interval in the construction with c1; c22 I gives us that

c1< a + b₂ < c2 and a + b₂ =2 C:

Thus there is a c = a+b

2 =2 C such that c1< c < c2. Hence

A = C \ [0; c) and E = C \ (c; 1]

are nonempty separated sets with A [ E = C. Thus C is totally disconnected and hence dimTC = 0 by Proposition 2.14.

The Cantor set is an invariant set. It has the similar contractions T1=x₃ and T2= 1 x₃;

i.e.

30 Chapter 6. Cantor sets

From (5.3) we thus have

dimSC =log 2_{log 3}:

The open set condition is fullled with O = (0; 1). Hence we have dimHC = dimPC = dimMC = dimSC = log 2_{log 3}

due to Proposition 5.11. From Proposition 5.3 we can now also verify that C is a fractal set. As a further exercise we calculate dimMC = log 2= log 3, thus

verifying the equality dimMC = dimSC.

Example 6.4. Looking at Figure 6.1, our starting interval C0 = [0; 1] only

needs one box of diameter 1 to cover it. Thus N(C0; 1) = 1. Next, we see

that C1 in its turn needs N(C1; 1=3) = 2 boxes to be covered, where 1=3 is the

scaling or similarity ratio. Following the pattern we have

N(C2; 1=9) = N(C2; 1=32) = 4 = 22; N(C3; 1=27) = N(C3; 1=33) = 8 = 23

and in general

N(Cn; (1=3)n) = 2n:

Thus

dimMC = lim_n!1log N(Cn; (1=3) n₎ log(1=(1=3)n₎ = lim n!1 log(2n₎ log(3n₎ = lim_n!1 n log 2 n log 3 = log 2 log 3: Proposition 6.5. If C is the ternary Cantor set we have

(C) = dimHC dimTC =log 2_{log 3} > 0

and C is a fractal set.

Proof. The proposition follows from Proposition 5.3 since dimHC = log 2= log 3

is non-integer.

We nish this chapter with an interesting theorem by Hausdor and then an interesting example of another Cantor set.

Proposition 6.6. (Theorem 6.6 in [21]) Every compact set is a continuous image of the Cantor set.

Proof. See p. 100 in [21].

Example 6.7. Let us now consider the Cartesian product of the set C(1=4) with itself, i.e. C(1=4)  C(1=4). Let us call it C02_{. From Proposition 6.1 we}

have that

dimHC0= log 2_{log 4} = 1₂;

an thus

6.2. Cantor set using ternary numbers 31

by Proposition 4.16. Now, C02 _{is obviously totally disconnected and thus}

dimTC02 = 0, which also can be seen by using Proposition 4.17 which gives

us

dimTC02  dimTC0+ dimTC0= 0;

since C0 _{is totally disconnected as well. And since C}02 _{is not the empty set we}

know that dimTC02 0. Hence the fractal degree of C02 is

(C02_{) = dim}

HC02 dimTC02= 1;

and thus a fractal set need not have an integer fractal degree value which one could have thought.

Chapter 7

The Sierpinski gasket

In this chapter we will start exploring the formulas in Section 4.2. We will however rst calculate the Minkowski dimension of a fractal set known as the Sierpinski gasket, S, with the aid of triangle shaped coverings. Consider a closed equilateral triangle, S0, of unit side. To obtain the Sierpinski gasket, start with

dividing S0 into four equally big triangles by joining the midpoint of each side

with one another. Now, remove the middle one of these, i.e. the open triangle containing the center of S0. The remaining set, S1, consists of three smaller

copies of the original one, now with the side length 1=2. Continuing with the same procedure with each of these three triangles leaves us with nine smaller equally big triangles and after that we have 27 smaller triangles and so on. Iterating further in innitely many steps nally gives us the limit set

S = \1

n=0

Sn;

known as the Sierpinski gasket as illustrated on next page.

A rst covering of S would of course be with a triangle of unit size. The next step is to cover S with three triangles of side "1 = 1=2, thus giving us

N(S; "1) = N(S; 1=2) = 3. Next we cover S with triangles of side "2 = "₂1 = 1₄

giving us a covering of S with nine triangles of side 1

4. Following the pattern

we have that S then needs to be covered with 27 triangles of side "3="₂2 = 1₈.

So in numbers we have: N(S; "0) = N(S; 1) = 1 N(S; "1) = N(S; 1=2) = 3 N(S; "2) = N(S; 1=4) = N(S; 1=22) = 9 = 32 N(S; "3) = N(S; 1=8) = N(S; 1=23) = 27 = 33 and in general N(S; "n) = N(S; (1=2)n) = 3n: Thus dimMS = lim_" n!0+ log N(S; "n) log(1="n) = limn!1 log N(S; "n) log(1="n) Leifsson, 2006. 33

34 Chapter 7. The Sierpinski gasket

Figure 7.1: The Sierpinski gasket = lim n!1 log N(S; (1=2)n₎ log(1=(1=2)n₎ = lim_n!1 log(3n₎ log(2n₎= lim_n!1 n log 3 n log 2= log 3 log 2; using (4.4).

Let us look at S from another point of view. Letting fc1; c2; c3g be the

vertices of S0, we dene the contractions Ti: Rn! Rn by

Ti(x) = 1₂(x ci) + ci; i = 1; 2; 3:

Thus

S = T1(S) [ T2(S) [ T3(S);

with the contraction ratio 1/2 for each Ti. Now, the Sierpinski gasket satises

the open set condition if we let O be the interior of the starting triangle, S0.

Hence, by (5.3), we have

dimSS = s = log 3_{log 2}:

Thus with reference to Proposition 5.11 we have

dimHS = dimPS = dimMS = dimSS =log 3_{log 2}:

Without thorough calculations, we also have the following from Proposition 4.16: dimH(S  I) = dimHS + dimHI = log 3_{log 2}+ 1:

since

dimHI = dimMI = 1:

Proposition 7.1. The topological dimension of the Sierpinski gasket is dimTS = 1:

35

Proof. To narrow it down we use Proposition 5.1 and conclude that dimTS  1

since dimHS = 1:584 : : :. We also know that S 6= ; so dimTS  0. Now, since

S is connected in its perimeter we have that dimTS 6= 0 by Proposition 2.14

and thus dimTS = 1.

Let us now sum this up with a proposition. Proposition 7.2. If S is the Sierpinski gasket then

(S) = dimHS dimTS = log 3_{log 2} 1 = 0:584 : : : ;

and thus S is a fractal set.

Proof. The proposition follows from Proposition 5.3 since dimHS is non-integer.

Example 7.3. Let us now develop further from what we have learned thus far in this chapter and verify the nite stability criterion for the Minkowski dimension. Let E = S [ I, where S is the Sierpinski gasket and I is a horizontal line segment of length 1, (it is helpful to think of I lying horizontal next to the base of S). We want to calculate dimME.

Our rst covering of E will consist of two squares of side 1, giving us N(E; 1) = 2 = 1 + 1:

Continuing like before we get:

N(E; 1=2) = 5 = 31_{+ 2}1_;

i.e. three squares to cover S and two squares to cover the line segment. N(E; 1=4) = 13 = 9 + 4 = 32_{+ 2}2_;

and in general:

N(E; (1=2)n_{) = 3}n_{+ 2}n

and thus

dimME = lim_n!1log N(E; (1=2) n₎ log(1=(1=2)n₎ = lim n!1 log(3n_{+ 2}n₎ log(2n₎ = lim n!1 log(3n_{(1 + (}2 3)n) log(2n₎ = lim n!1 log(3n_{) + log(1 + (}2 3)n) log(2n₎ = lim n!1 log(3n₎ log(2n₎ = log 3_{log 2} = maxflog 3 log 2; 1g

36 Chapter 7. The Sierpinski gasket

in accordance with (III') in Section 1.3.

7.1 The Sierpinski gasket using the ternary tree

With the knowledge of how the ternary number system works from before, we can now consider a simple procedure which iterates to the Sierpinski gasket, S. What we will consider is a so called ternary tree. Starting at a point given the value 0, we draw three outgoing lines of equal length from this point and with an angular distance of 120_{. One line is drawn to the right, one line}

to the left upwards and one line to the left downwards. Thereafter we label them with 0, 1 and 2 respectively. Then, if we continue to draw lines in the same manner outwards from the end of each of these lines and adding the number corresponding to its direction in the end of its value gives us all of the ternary numbers in [0; 1]. Continuing like this indenitely, we receive a structure agreeing well with our Sierpinski gasket. Actually, the labeling of numbers in this case is not so important as long as we make homogeneous iterations all the time. This is best shown with an illustration.

Figure 7.2: The ternary tree approximation of the Sierpinski gasket.

7.2 The Sierpinski sieve

Let us look at yet another approach to construct the Sierpinski gasket. We now construct the Sierpinski gasket using modulo 2 arithmetic on Pascal's triangle. An illustration of the rst rows of Pascal's triangle follows below. Each number inside Pascal's triangle is the sum of the two numbers above it, i.e 6 = 3 + 3, 15 = 5 + 10 and so on. One can now choose between many dierent approaches

7.2. The Sierpinski sieve 37

Figure 7.3: The rst rows of Pascal's triangle.

to construct the Sierpinski gasket from this triangle. We will use modulo 2 arithmetic, i.e. considering even and odd numbers inside Pascal's triangle. Now apply modulo 2 arithmetic on Pascal's triangle in the sense that we color each position in the triangle white if it consists of an even number, and each position black if it contains an odd number. Iterating this process throughout Pascal's triangle leaves a good approximation to the Sierpinski gasket known as the Sierpinski sieve, illustrated below.

Figure 7.4: The rst rows of the Sierpinski sieve.

Looking at the rst two rows of the colored Pascal's triangle we see that this corresponds to the rst approximation, S1, of the Sierpinski gasket. Adding

another two rows we notice that these four rows correspond to S2. Generalizing

this we have the rst 2k _{colored rows correspond to S} k.

Chapter 8

The von Koch snow ake

We will now consider another fractal set called the von Koch snow ake, illus-trated below.

Figure 8.1: The von Koch snow ake.

The von Koch snow ake consists of three congruent fractals K, called von Koch curves. It is therefore enough to only study K in order to derive the properties of the von Koch snow ake.

40 Chapter 8. The von Koch snow ake

To construct K, start with a line segment and call this K0. Now, we remove

the middle third of K0 and replace it with the upright sides of an equilateral

triangle, so each of these four line segments are of equal length. Call this curve K1. Next, we repeat this procedure on each of the four segments of K1, giving

us K2 consisting of 42= 16 line segments of equal length. Thus K3consists of

43 _{= 64 line segments and so on. K is now dened as the limit set obtained}

after innitely many iterations, i.e. K = lim

n!1Kn:

We now look at one of the properties of K and hence the von Koch snow ake thereby.

Proposition 8.1. The von Koch curve is of innite length.

Proof. Let K0have length 1. Then K1has length 4=3, K2has length 42=32and

so on. Thus Kn constructed after n steps is of length 4n=3n. Hence intuitively

K should have length

lim

n!1

4n

3n = 1:

For a rigorous proof, we take an advance look at Proposition 8.3 below and notice that dimHK > 1. Thus using Remark 3.10 we have that K's length is

in fact innite.

Corollary 8.2. The von Koch snow ake has innite length.

The von Koch curve is obviously self-similar, while the von Koch snow ake is not since it has no copies of itself in its structure. The open set condition is fullled with O being the open equilateral triangle with side length equal to K0.

K can be dened through the following contractions using complex numbers: T1= (1₂+ p 3i 6 )z and T2= (1₂ p 3i 6 )(z 1) + 1:

Thus K is an invariant set under these contractions. We can now conclude that the Moran equation (5.2) is satised with c = j1

2 + p 3i 6 j = 1= p 3 and m = 2, giving us

dimSK =log 4_{log 3};

i.e. twice as big as the similarity dimension of the Cantor dust. Proposition 8.3. Let K be the von Koch curve. Then

dimHK = dimPK = dimMK = dimSK = log 4_{log 3}:

Proof. The proposition follows from our recent arguing above and from Propo-sition 5.11.

41

Figure 8.2: The rst two coverings of K.

Example 8.4. We will calculate the Minkowski dimension of K using square box coverings. In our rst cover we use three boxes to cover K. Thus we have "1 = 1=3 and N(K; "1) = 3. In the next step we use 12 covers and

"2= "1=3 = 1=9. This is illustrated in a simple gure below.

All in all we have

N(K; "1) = N(K; 1=3) = 3, N(K; "2) = N(K; 1=9) = N(K; 1=32) = 12 = 3
4, N(K; "3) = N(K; 1=27) = N(K; 1=33) = 48 = 3
42, and in general N(K; "n) = N(K; 1=3n) = 3
4n 1. Hence dimMK = lim_" n!0 log N(K; "n) log(1="n) = limn!1 log N(K; "n) log(1="n) = lim n!1 log N(K; 1=3n₎ log(1=(1=3)n₎ = lim_n!1 log(3
4n 1₎ log(3n₎ = lim n!1 (n 1)log 4 + log 3 n log 3  = log 4_{log 3}: Proposition 8.5. If K is the von Koch curve then dimTK = 1.

Proof. First of all, since dimHK = 1:261 : : : we know that dimTK  1 by

Proposition 5.1. And since K is not the empty set we can also conclude that dimTK  0. From Proposition 2.14 we have that dimTK 6= 0 because K is

connected.

Proposition 8.6. The von Koch curve K is a fractal set with the fractal degree (K) = log 4_{log 3} 1:

42 Chapter 8. The von Koch snow ake

Proof. From Proposition 5.3 we have that K is a fractal since dimHK is

non-integer and remembering Denition 5.2 we have that the fractal degree of K is

(K) = dimHK dimTK =log 4_{log 3} 1 = 0:2618 : : : :

8.1 The von Koch curve versus C

We now brie y compare the von Koch curve, K, with the Cantor product C2₌

C  C, where C is the middle third Cantor set.

Figure 8.3: The Cantor product C2_.

With the dimension formulas in Section (4.2) we directly have

dimH(Cn) = dimP(Cn) = dimM(Cn) = dimS(Cn) = n dimHC = nlog 2_{log 3}

(8.1) where Cn<sub>= C  C 


 C</sub> | {z } n times  Rn_:

Now, from (8.1) we have that

dimHC2= 2log 2_{log 3};

which leads us to the rst interesting observation that dimHC2= dimHK:

Also, noticing that C2 _{is totally disconnected whilst K is not we know that}

Chapter 9

Space-lling curves

Before digging into any examples of space-lling curves we shall dene its con-cept.

Denition 9.1. A continuous function f : [0; 1] ! Rn_{, n  2, is called a}

space-lling curve if the n-dimensional Lebesgue measure, Ln_{, of its direct image,}

f= f([0; 1]), is strictly positive, i.e. Ln(f) > 0.

Remark 9.2. The mapping under a space-lling curve, S, from [0; 1] to the space that S lls is surjective, however not injective. (See Netto's theorem in e.g. [21]).

9.1 Peano space-lling curve

We shall now consider a fractal curve called the Peano space-lling curve, P , which maps the closed unit interval onto the closed unit square i.e. it lls the unit square. It was Giuseppe Peano who rst discovered a curve of this type. His approach was purely analytic. The geometric approach we will use here was deduced by David Hilbert one year later, therefore this Peano curve is also known as the Hilbert curve. In fact all space-lling curves are called Peano curves.

Our starting set consists of the closed unit square [0; 1]2_{. To generate the}

rst approximation P1of P we think of the unit square as a partition into four

connected squares of side 1/2. Now, starting at t = 0 a line is drawn into the center of the rst square then leading on to the center of the next square so that the centers of all four squares are visited only once, and the endpoint of the line is in t = 1. This is illustrated below. The thinner lines are not part of the curve. To generate P2 we now think of the unit square as 16 = 42 connected squares

of side 1/4. The procedure is now the same, we want to pass through every center of the squares once, starting at t = 0 and nishing at t = 1. Following the pattern we have that Pn can be generated if we think of the unit square as

4n _{squares of side 1=2}n _{through which we want to draw a polygon curve similar}

to the one before, i.e. passing through the center of each subsquare. A simple illustration of P2 to P5follows below.

The length of Pn is obviously

2n 1

2n;

44 Chapter 9. Space-lling curves

Figure 9.1: The rst iteration, P1, of the Hilbert curve.

thus

P = lim

n!1Pn

intuitively has innite length. Referring to Proposition 9.3 and Remark 3.10 as we have done before, we can conrm that this is true. Let us show that it really is a mapping from [0; 1] to [0; 1]2_{. Looking at the gures we can with a simple}

reasoning see that

jPn+1(t) Pn(t)j 

p 2

2n for t 2 [0; 1]:

Thus for m > n we have that sup 0t1jPm(t) Pn(t)j  sup0t1 m 1_X i=n jPi+1(t) Pi(t)j X1 i=n p 2 2i = 2
p 2 2n ! 0; as m; n ! 1:

Thus (Pn)1n=1 is uniformly convergent according to the Cauchy criterion for

uniform convergence, which states that a sequence, (Pn)1n=1, of functions dened

on a set E  Rn _{is uniformly convergent if and only if}

sup

t2EjPm(t) Pn(t)j ! 0 as m; n ! 1:

Now, P is continuous on [0; 1] due to the fact that if a sequence of continuous functions converges uniformly towards a function P on an interval, then P is continuous (see 2.1.8. in [18]). We also have that [0; 1] is compact, and so P ([0; 1]) is compact. Thus, since every point in [0; 1]2_{is an accumulation point}

of P ([0; 1]), we have that

P ([0; 1]) = [0; 1]2_:

Proposition 9.3. The Hilbert curve, P , is a space-lling curve.

Proof. From our calculations above we get that the direct image of P ([0; 1]) is [0; 1]2_{. Since L}2_{([0; 1]}2_{) = 1 > 0 the claim thus follows.}

From (5.3), with m = 4 and c = 1=2, we get dimSP = log 4_{log 2} = 2:

9.2. The Heighway dragon 45

Figure 9.2: The second to fth iterations of the Hilbert curve. Remembering Remarks 3.8 and 3.10 we also notice that dimHP = 2 and so

dimHP = dimPP = dimMP = dimSP = 2;

in accordance with Remark 5.12 and Proposition 4.14. The topological dimen-sion of the Hilbert curve, or Peano curve, is 2 since it lls the plane. Hence the fractal degree of P is

(P ) = dimHP dimTP = 2 2 = 0:

Thus our denition of a fractal set tells us that The Hilbert curve is not a fractal set though we intuitively know it is. Fractals with this property are sometimes referred to as borderline fractals.

9.2 The Heighway dragon

Let us take look at another space-lling curve called the Heighway dragon after its founder John E. Heighway. It can be constructed in many dierent ways. We will use line segments to construct it. Our starting set, D0, consists of the

unit interval. To create D1we replace D0with two line segments each of length

d1 = 1=p2 joined at a right angle. Next we receive D2 by replacing each line

segment in D1 with two line segments each of length