Examensarbete

Fractal sets and dimensions Patrik Leifsson

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

Fractal sets and dimensions

Applied Mathematics, LinkopingsUniversitet

Patrik Leifsson

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

Examensarbete: 20 p

Level: D

Supervisor: Jana Bjorn, Applied Mathematics, LinkopingsUniversitet

Examiner: Jana Bjorn, Applied Mathematics, LinkopingsUniversitet

Linkoping: May 2006

Datum Avdelning, Institution Date Division, Department

Matematiska Institutionen May 2006 581 83 LINKOPING SWEDEN

Sprak Rapporttyp ISBN Language Report category ISRN Svenska/Swedish Licentiatavhandling LiTH - MAT - EX - - 06 / 06 - - SE x Engelska/English x Examensarbete C-uppsats Serietitel och serienummer ISSN D-uppsats Title of series, numbering 0348-2960 Ovrig rapport

URL forelektronisk version http://www.ep.liu.se/exjobb/mai/2006/tm/006/

Titel Fractal sets and dimensions Title

Forfattare Patrik Leifsson Author

Sammanfattning 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 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 di erent fractals are studied and compared. A key idea in this thesis has been to sum up di erent names and de nitions referring to similar concepts.

Nyckelord Keyword 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. vi 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 di erent fractals are studied and compared. A key idea in this thesis has been to sum up di erent 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.

Leifsson, 2006. vii viii Acknowledgements

I would like to thank my supervisor and examiner Jana Bjornfor 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.

Leifsson, 2006. ix x

Preliminaries

In this section we collect some basic notations and de nitions.

x x  y means that there exists c > 0 such that 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.

n Ball For e0 2 R and R 3 " > 0, we de ne an open ball, B, as

n B(e0; ") = fe 2 R : je e0j < "g; and a closed ball, B, as

n B(e0; ") = fe 2 R : 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  Rn 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. n n Closure of a set For a set E  R and a point x0 2 R 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: 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

Leifsson, 2006. xi xii Contents 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 di erent 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

X1 3 f(x) = ai cos(bix); 0 < a < 1; ab > 1 + ; 2 i=0 which is continuous everywhere but nowhere di erentiable. 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 di erent 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 di erent types of dimensions and measures associated with them. In Chapter 4 we also extend our dimension algebra with some product relations

Leifsson, 2006. 1 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 di erent 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 di erent 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: 8 < (i) dimfag = 0; where fag is the singleton set. (I) (ii) dim I1 = 1; where I1 is the unit interval. : (iii) dim Im = m; where Im is the m-dimensional hypercube.

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

1 n (III) Countable stability: If fAigi=1 is a sequence of closed subsets of R , then [1  dim Ai = sup dim Ai: i=1 i1

n (III') Finite stability: If A1;A2;:::;Am are closed subsets of R , then [m  dim Ai = max dim Ai: 1im i=1

(IV) Invariance: If  : Rn ! Rn 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 di erent 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 dimT E.

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.

Leifsson, 2006. 3 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.

n n De nitionS 2.4. A family F of subsets of R is a cover of a set E  R if E  F . F 2F 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  Rn 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 vertical list and then, starting at (a1; b1) move successively through the list of 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-Bernsteintheorem [3] p. 100), Q2 = Q ¢ Q is countable, and by induction, Qn is countable for all n. Thus Rn is separable since Rn = Qn 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

dimT E = ind E: De nition 2.13. The set E is said to be totally disconnected if for any e1; e2 2 E, e1 6= 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 dimT E = 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 dimT fag = 0 by Proposition 2.14. (I)(ii) The open intervals

1 Ix;" = (x "; x + "); x 2 I = I; " > 0; constitute a topological base for the interval I. Now since

dimT @Ix;" = dimT fx "; x + "g = 0 by (i), we have that dimT I  1 and the fact that I is connected gives us dimT I 6= 0 and thus dimT I = 1. m (I)(iii) That dimT I = m can be shown by an argument similar to that for (I)(ii). n (II) Let A  E  R and dimT E = 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

dimT (@(A \ U) \ A) < dimT @U  k 1; 6 Chapter 2. The topological dimension

we obtain dimT A  k = dimT E. (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 U 0 = 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

dimT f(E)  dimT E; 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 q1 and q2 both 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 dimT Q = 0 by Proposition 2.14. In an analogous manner with the same references it follows that dimT I = 0: So we have that dimT I = dimT Q = 0; but dimT (I [ Q) = dimT R = 1 6= maxfdimT Q; dimT Ig: Proposition 2.18. (Theorem 3.2.10 in [7]) If A; E  Rn, then

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

Example 2.17 shows that this connection cannot be improved.

n Proposition 2.19. (Theorem A.4.14 in [6]) dimT R = n. Remark 2.20. In this chapter we have seen examples of di erent methods to calculate a set's topological dimension. The key idea to calculate dimT E 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 S1 1 De nition 3.1. We say that fEigi=1 is a -cover of a set E if E  Ei and i=1 n 0  diam Ei  , for all i. For a set E  R , s  0, and  > 0, de ne X1 s s H(E) = inf (diam Ei) ; i=1

1 where the in mum are taken over all (countable) -covers fEigi=1 of E. Also de ne the s-dimensional Hausdor measure by

s s s H (E) = lim H(E) = sup H(E): !0 >0 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  X1  Ei  (Ei) i=1 i=1

n for all Ei  R . s Proposition 3.3. H 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

s  H(;) = 0 since diam ; = 0 . n 1  Let A  E  R and " > 0. Then there is a -cover fEig1=1 of E such that X1 s s (diam Ei)  H(E) + ": i=1 1 But fEig1=1 is also a valid -cover of A and thus X1 s s s H(A)  (diam Ei)  H(E) + ": i=1 Now, letting " ! 0 gives us

s s H(A)  H(E):

P1 s  Assuming H(Ei) < 1 we have for an arbitrary " > 0 that for each i=1 i 1 i  1 there exists a -cover fAjgj=1 of Ei such that X1 i s s i (diam Aj) < H(Ei) + 2 ": j=1

S1 i Thus fAjgi;j1 constitutes a valid -cover of Ei and i=1 X X1   X1 i s s i s (diam Aj)  H(Ei) + 2 " = H(Ei) + ": i;j1 i=1 i=1 Letting " ! 0 proves the claim.

s 1 Proposition 3.4. H is a measure, i.e. if fAigi=1 is a pairwise disjoint count- able collection of measurable sets, then [1  X1  Ai = (Ai): (3.1) i=1 i=1 Proof. See for example Theorem 4.2 in [16].

s Remark 3.5. H 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 X1 1 j 1 H (Ai) = inf (diam Ei ) = 1; (3.2) j=1 3.2. The Hausdor dimension 9

j j where Ei are -covers of Ai with diam Ei  . Since we are considering Carte- sian products fqig ¢ [0; 1] we have X1 j (diam Ei )  1 j=1 and thus (3.2) is ful lled. Then using balls, Bi, of radii  to cover E we need 1 approximately c ¡ 2 of these, where c 2 R. Thus [1  X1 1 1 X1 H1 A = inf (diam B )  c ¡ ¡   < H1(A ) = 1:  i i 2   i i=1 i=1 i=1

s Hence the necessary condition (3.1) is not ful lled in order for H to be a measure. De nition 3.7. The Lebesgue measure Ln on Rn is de ned as follows. For A of the form n A = f(x1; : : : ; xn) 2 R : ai  xi  big (3.3) de ne n L (A) = (b1 a1)(b2 a2) ¡¡¡ (bn an); and extend Ln to general subsets of Rn by nX1 [1 o n n L (A) = inf L (Ai): A  Ai for all Ai of the form (3:3): i=1 i=1 Remark 3.8. It can be proved that when s = n on Rn, then Hn 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

s s dimH (A) = supfs : H (E) > 0g = supfs : H (E) = 1g = infft : Ht(E) < 1g = infft : Ht(E) = 0g:

Remark 3.10. The previous de nition can also be expressed as  1 if s< dim (E); Hs(E) = H 0 if s> dimH (E);

s and H (E) can attain any value in [0; 1] for s = dimH (E).

In other words, dimH E 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 for dimensions outlined in Section 1.3. 10 Chapter 3. The Hausdor measure and dimension

Proof. We again treat each condition separately. (I)(i) We have dimH fag = 0 for the singleton set fag, since diamfag = 0 gives us

Hs(fag)  (diamfag)s = 0 for all s > 0, and thus

s dimH fag = inffs : H (fag) = 0g = 0:

1 1 1 (I)(ii) We have 0 < H (I ) < 1 and hence dimH I = 1, using Remark 3.10.

(I)(iii) Let Im be an m-dimensional hypercube in Rn, 1  m  n. We have

Hm(Im) = cLm(Im) 2 (0; 1):

m Hence dimH I = m, using Remark 3.10.

(II) Monotonicity: Let A  E  Rn. Since Hs(A)  Hs(E) when n A  E  R , it follows from De nition 3.9 that dimH (A)  dimH (E).

(III) Countable stability: From the monotonicity we have for all j,

[1  dimH Ei  dimH Ej i=1 and hence, taking supremum over all j,

 [1  dimH Ej  sup dimH Ej: j=1 j1

The inequality [1  dimH Ei  sup dimH Ej i=1 j1 s follows from the fact that if s > dimH Ei for all i, then H (Ei) = 0  S1  s for all i, which gives us H Ei = 0 and hence i=1 [1  dimH Ei  s: i=1

(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 log 4 has Hausdor dimension log 3 is 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]).

Proposition 3.13. If E is a nite or countable set, then dimH (E) = 0. 3.2. The Hausdor dimension 11

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

0 H (Ei) = 1:

Thus dimH (Ei) = 0: Hence, by the countable stability

[1  dimH Ei = 0: i=1

This is a usefull proposition which we will take advantage of later on. 12 Chapter 3. The Hausdor measure and dimension 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 dimension, 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

s dimM E = inffs : lim sup N(E;")" = 0g; "!0+ 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

s dimM E = inffs : lim inf N(E;")" = 0g: "!0+

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 xi 2 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;"), xi 2 E, i = 1; : : : ; k. Sk Now, if there exists an x in E n i=1 B(xi; 2") then the balls B(x1;");:::;B(xk;");B(x; ") are pairwise disjoint and thus

k + 1  P (E;") = k

Leifsson, 2006. 13 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);

n let k1 = N(E; "=2) and k2 = P (E;") and let x1; : : : ; xk1 2 R and y ; : : : ; y 2 E be such that 1 k2

[k1 E  B(xi; "=2) i=1 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

log N(E;") log P (E;") dimM E = lim sup = lim sup (4.2) "!0+ log(1=") "!0+ log(1=") and log N(E;") log P (E;") dimM E = lim inf = lim inf : (4.3) "!0+ log(1=") "!0+ log(1=")

Let us show the rst equality in (4.2). The second equality in (4.2) follows from Proposition 4.2. Suppose s > dimM E. Then for all  there exists an "0 > 0 such that s N(E;")" < ; for all " < "0: Then log N(E;")"s < log  so log  log N(E;") s > ; log " and letting " ! 0+ and taking in mum over all s > dimM E, we get log N(E;") dimM E  lim sup : "!0+ log(1=") Now suppose log N(E;") lim sup < s < dimM E: "!0+ log(1=")

Then there exist "j ! 0 and 0 > 0 such that

s N(E;"j)"j > 0 > 0 15 which implies

log N(E;"j) + s log "j > log 0: But then log N(E;" ) log  log N(E;") s  lim sup j 0  lim sup ; "j !0+ log(1="j) "!0+ log(1=") which is a contradiction. Thus log N(E;") dimM E = lim sup : "!0+ log(1=")

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

log N(E;") log P (E;") dimM E = lim = lim : (4.4) "!0 log(1=") "!0 log(1=")

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 n volume of an "-neighbourhood of E  R . The "-neighbourhood, E", of E is de ned as n E" = fx 2 R : 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

n log L (E") dimM E = n + lim sup ; (4.6) "!0+ log(1=")

n log L (E") dimM E = n + lim inf ; (4.7) "!0+ log(1=") and n log L (E") dimM E = n + lim (4.8) "!0+ log(1=") if it exists. 16 Chapter 4. Minkowski dimensions

Proof. First we prove that the inequalities

n n n c" P (E;")  L (E")  c(2") N(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 n n L (E")  c(2") N(E;"): To understand that n n c" P (E;")  L (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"nP (E;") thereby. Now, using (4.9), we have

log P (E;") log(c 1" nLn(E )) dim E = lim sup  lim sup " M log(1=") log(1=") "!0+  "!0+  log Ln(E ) log c n log " = lim sup " log " "!0+   log Ln(E ) = lim sup " + n "!0+ log " log Ln(E ) = n lim inf " : "!0+ log "

n n This was attained from c" P (E;")  L (E"). Using the last inequality in (4.9) we have

log N(E;") log(c 1(2") nLn(E )) dim E = lim sup  lim sup " M log(1=") log(1=") "!0+  "!0+  log Ln(E ) log c n log(2") = lim sup " log " "!0+   log Ln(E ) = lim sup " + n "!0+ log " log Ln(E ) = n lim inf " : "!0+ log "

Thus we conclude that

n log L (E") dimM E = n lim inf ; "!0+ log " which proves (4.6). Now, (4.7) follows in an manner analogous to the above. Finally, n log L (E") dimM E = n + lim "!0+ log(1=") follows from (4.6) and (4.7). 17

From the de nition of the Hausdor measure we can deduce the useful re- s s 1 lation H(E)  N(E; ) . In other words, if fAigi=1 is a -cover of E, then as diam Ai   for all i, we have X1 (diam A )s  s + s + ¡¡¡ + s = N(E; )s: i | {z } i=1 N(E;) Theorem 4.4. For all E  Rn,

dimH E  dimM E  dimM E: (4.10)

Proof. If s < dimH E, then

s s s 0 < H (E) = lim H(E)  lim N(E; ) !0+ !0+ and thus log N(E; ) + s log  > log Hs(E) 1, for small enough  > 0. From this we have log N(E; ) log Hs(E) s  lim inf = dimM E  dimM E: !0+ log(1=) 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 1 1 1 to its Minkowski dimension. The set E = f0; 1; 2 ; 3 ; 4 ;:::g has Hausdor di- 1 mension 0 since it is countable. However, dimM E = 2 . To show this, let " > 0 and k be the smallest integer such that 1 1 1 = < ": k 1 k k(k 1)

1 A rst-order approximation of " in terms of k is "  k2 . The number of balls of radius " it takes to cover the points 1; 1 ; 1 ; 1 ;:::; 1 equals k 1  p1 . And 2 3 4 k 1 " 1 to cover the points which lie in E \ [0; k ] by balls of radius ", it takes about 1  p1 balls. Hence the number of balls needed to cover E is essentially 2k" 2 " 1 1 1 N(E;")  p + p  p 2 " " " from which we obtain

log( p1 ) 1 log N(E;") " 2 log " 1 dimM E = lim = lim = lim = : "!0+ log " "!0+ log " "!0+ log " 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, dimM E, ful lls conditions (I), (II) and (IV') of these. 18 Chapter 4. Minkowski dimensions

Proof. (I)(i) We have

s dimM fag = inffs : lim sup N(fag;")" = 0g: "!0+

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

lim sup N(fag;")"s = 0 "!0+ and hence 0  dimM fag  dimM fag = 0. (I)(ii)-(iii)We have N(Im;")  " m and hence

m m m m log N(I ;") dimM I = dimM I = dimM I = lim = m: "!0+ log(1=")

(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

dimM A  dimM E and dimM A  dimM E and so dimM A  dimM E by (4.2){ (4.4). 1 (III') Let fEijgj=1 be a -cover of Ai for i = 1; : : : ; n. Then

fEij; i = 1; : : : ; n; j = 1;:::; 1g

Sn is a -cover of i=1 Ai, so [n  Xn N Ai;   N(Ai; ): i=1 i=1

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

s lim sup N(Ai; ) = 0: !0+ Adding over i = 1; : : : ; n gives us [n  s lim sup N Ai;   = 0: !0+ i=1

Sn Thus from De nition 4.1 we have dimM ( i=1 Ai)  s. Hence [n dimM ( Ai)  max dimM Ai: i=1;:::;n i=1

Sn Now, suppose that dimM ( Ai) < max dimM Ai, i.e. there is an i such i=1 i=1;:::;n that [n  dimM Ai > dimM Ai : i=1 4.1. The packing dimension 19

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

dimM g(E)  dimM E and dimM g(E)  dimM E; 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 1 1 1 and Minkowski dimensions. The set to consider is F = f0; 1; 2 ; 4 ; 8 ;:::g. Since F  E, where E is the set considered in Example 4.5, we immediately know 1 that dimM F  dimM E = 2 from the monotonicity condition. We also know n that dimH F = 0 since F is countable as well. Now, with 0 <  = 2 we we get a valid -cover of F by N(F; 2 n) = n + 2 balls. Since

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

dimM F = dimM F = 0: Now consider the mapping

1 1 1  : x ! x ; 2 from E to F for each nonzero element x in E. Since

1 1 1 lim x = 0; x!0 2 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 dimM . 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 n o n o dimP E = inf sup dimM Ei and dimP E = inf sup dimM Ei ; i1 i1

1 n where the in ma are taken over all countable covers fEigi=1 of E  R by bounded sets.

With this de nition, we get dimP E = dimP E = 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 X1 s s s s s P" (E) = sup (diam Bi) and P (E) = lim P" (E) = inf P" (E): "!0+ ">0 i=1

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 nX1 [1 o s s P (E) = inf P (Ei): E = Ei : i=1 i=1 Proposition 4.11. Ps(E) is a measure on Rn. 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,

s s dimP E = inffs : P (E) = 0g = inffs : P (E) < 1g = supfs : Ps(E) > 0g = supfs : Ps(E) = 1g:

Proof. See Theorem 5.11 in [16]. De nition 4.13. For E  Rn, we de ne the packing dimension as

dimP E = dimP E:

Proposition 4.14. If E  Rn then

dimT E  dimH E  dimP E  dimM E:

Proof. The rst inequality follows from Proposition 5.1. The second inequality, i.e. dimH E  dimP E, follows from the fact that

Hs(E)  Ps(E) for all E  Rn;

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

dimH E  dimP E 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 < dimP E we have that

P s(E)  Ps(E) = 1:

1 So for 0 <   1 there are disjoint balls fBigi=1 with centers in E and radii at most equal to  such that

X1 s 1 < (diam Bi) : i=1

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

k 1 k 2 < diam Bi  2 :

Then X1 ks 1 < nk2 (4.11) k=0 is also satis ed. Now, unless we want (4.11) contradicted there need to be some k with kt t s nk > 2 (1 2 ) 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 < . k 1 Hence P (E; 2 )  nk and

k 1 t k 1 k 1 t t t s (2 ) ¡ P (E; 2 )  nk(2 ) > 2 (1 2 ); where 2 k 1 < . Thus

lim sup P (E; )t  2 t(1 2t s) > 0 !0 so dimM E  t, for all t < dimP E 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) = dimM E; for all open sets G intersecting E, then

dimP E = dimM E:

4.2 Product relations

There are some valuable formulas which can reduce the amount of e ort 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)  dimH A + dimH E; (4.12)

dimH (A ¢ E)  dimH A + dimM E; (4.13)

dimM (A ¢ E)  dimM A + dimM E; (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  Rn be two sets, not both empty. Then

dimT (A ¢ E)  dimT A + dimT E: Chapter 5

Fractals and self-similarity

5.1 Fractals

Proposition 5.1. ([12], p. 3) For any set E we have dimT E  dimH E. A complete proof will not be given. Some things can be noted however. For E = ; we obviously have dimT E < dimH E since dimT ; = 1 and dimH E  0. Remembering Proposition 3.13 we know that for a countable set E we have dimH E = 0. Referring to Proposition 2.14, which says that a compact totally n disconnected set E  R has dimT E = 0, we have these cases covered too. The cases left to study we leave unproven. (See e.g. p. 104 in [13]).

n De nition 5.2. We say that a set E  R is fractal if dimT E < dimH E. The fractal degree of the set E is (E) = dimH E dimT E.

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.

Leifsson, 2006. 23 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 [m E = Ti(E): i=1 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 Ti(O)  O i=1 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, dimS E, is the unique solution s to the Moran equation

s s s c1 + c2 + ¡¡¡ + cm = 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 log m dim E = s = : (5.3) S log(1=c) 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- n ilarities on R satisfying the open set condition with contraction ratios ci, m i = 1; 2; : : : ; m. If E is the invariant set of fTigi=1, then

 dimH E = dimP E = dimM E = dimS E;

s s  0 < H (E) < 1 and P (E) < 1, where s = dimS E;

 There exist e1; e2 2 (0; 1) such that for s = dimS E,

s s s e1r  H (E \ B(x; r))  e2r

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

dimH E = dimP E = dimM E  dimS E:

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

Proposition 5.13. (Proposition 9.6 in [9]). For contractions T1;T2;:::;Tm n with contraction ratios ci < 1; i = 1; : : : ; m, on a closed invariant set E  R we have that dimH E  s and dimM E  s; where (5.2) is ful lled. 26 Chapter 5. Fractals and self-similarity Chapter 6

Cantor sets

1 Generally, for 0 < < 2 , we de ne Cantor sets on R as the limit set \1 [2i C( ) = Ei;j i=0 j=1 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 Ei;1;:::;Ei;2i are de ned through removing i 1 intervals of length (1 2 ) diam Ei 1;j = (1 2 ) from the middle of each i interval Ei 1;j. Thus each Ei;j has length . 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.

log 2  dimH C( ) = log(1= ) . s  H (C( )) = 1, where s = dimH C( ).

6.1 The ternary Cantor set

Among the di erent 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 \1 [2i 1 1 2 C( 3 ) = Ei;j; where E0;1 = [0; 1];E1;1 = [0; 3 ];E1;2 = [ 3 ; 1] i=0 j=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 C1 we remove the open middle third interval from C0, i.e.

1 2 1 2 C1 = C0 n ( 3 ; 3 ) = [0; 3 ] [ [ 3 ; 1]:

Leifsson, 2006. 27 28 Chapter 6. Cantor sets

From each of these two intervals we then remove the open middle third intervals of length  112 1 1 1 2 ¡ = : 3 3 9 Thus 1 2 7 8 1 2 1 2 7 8 C2 = C1 n (( 9 ; 9 ) [ ( 9 ; 9 )) = [0; 9 ] [ [ 9 ; 3 ] [ [ 3 ; 9 ] [ [ 9 ; 1]: Continuing like this in in nitely many steps gives us our limit set \1 C = Ci: i=0 This set is obviously quite porous containing no intervals of positive length, hence its name Cantor dust. A simple illustration of the rst di erent 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 di erent 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 + 21 + 20 3 so we have 710 = 1112. Another example is 810 = 2 = 10002. In an analogous manner we can rewrite any fraction written in base n into another base m 6= n. E.g. 45 1 1 1 1 = + + + = 0:101101 : 64 2 8 16 64 2 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,

1 0 1 0 710 = 2 ¡ 3 + 3 = 213 and 810 = 2 ¡ 3 + 2 ¡ 3 = 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 di erent expansions, e.g. 1 = 0:1000::: = 0:0222::: ; 3 3 3 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.

1 Proof. Consider the second generation C1 of C and the binary sequences (xi)i=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 1 sequences (xi)i=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 T1 closed since in C = i=0 Cn each Cn consists 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; c2 2 C and c1 < c2. Let n  = c2 c1. Each interval Cn 2 C is of length 3 . Choosing n such that n 3 <  places c1 and c2 in di erent intervals. Supposing I = [a; b] is the last interval in the construction with c1; c2 2 I gives us that a + b a + b c < < c and 2= C: 1 2 2 2

a+b Thus there is a c = 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 dimT C = 0 by Proposition 2.14. The Cantor set is an invariant set. It has the similar contractions x x T = and T = 1 ; 1 3 2 3 i.e. C = T1(C) [ T2(C): 30 Chapter 6. Cantor sets

From (5.3) we thus have log 2 dim C = : S log 3 The open set condition is ful lled with O = (0; 1). Hence we have log 2 dim C = dim C = dim C = dim C = H P M S 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 dimM C = log 2= log 3, thus verifying the equality dimM C = dimS C.

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

2 2 3 3 N(C2; 1=9) = N(C2; 1=3 ) = 4 = 2 ;N(C3; 1=27) = N(C3; 1=3 ) = 8 = 2 and in general n n N(Cn; (1=3) ) = 2 : Thus

n log N(Cn; (1=3) ) dimM C = lim n!1 log(1=(1=3)n) log(2n) n log 2 log 2 = lim = lim = : n!1 log(3n) n!1 n log 3 log 3 Proposition 6.5. If C is the ternary Cantor set we have log 2 (C) = dim C dim C = > 0 H T log 3 and C is a fractal set.

Proof. The proposition follows from Proposition 5.3 since dimH C = 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 log 2 1 dim C0 = = ; H log 4 2 an thus 1 dim C02 = 2 ¡ H 2 6.2. Cantor set using ternary numbers 31 by Proposition 4.16. Now, C02 is obviously totally disconnected and thus 02 dimT C = 0, which also can be seen by using Proposition 4.17 which gives us 02 0 0 dimT C  dimT C + dimT C = 0; since C0 is totally disconnected as well. And since C02 is not the empty set we 02 02 know that dimT C  0. Hence the fractal degree of C is

02 02 02 (C ) = dimH C dimT C = 1; and thus a fractal set need not have an integer fractal degree value which one could have thought. 32 Chapter 6. Cantor sets 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

\1 S = Sn; n=0 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 "1 1 N(S; "1) = N(S; 1=2) = 3. Next we cover S with triangles of side "2 = 2 = 4 1 giving us a covering of S with nine triangles of side 4 . Following the pattern "2 1 we have that S then needs to be covered with 27 triangles of side "3 = 2 = 8 . So in numbers we have:

N(S; "0) = N(S; 1) = 1

N(S; "1) = N(S; 1=2) = 3 2 2 N(S; "2) = N(S; 1=4) = N(S; 1=2 ) = 9 = 3 3 3 N(S; "3) = N(S; 1=8) = N(S; 1=2 ) = 27 = 3 and in general n n N(S; "n) = N(S; (1=2) ) = 3 : Thus log N(S; "n) log N(S; "n) dimM S = lim = lim "n!0+ log(1="n) n!1 log(1="n)

Leifsson, 2006. 33 34 Chapter 7. The Sierpinski gasket

Figure 7.1: The Sierpinski gasket

log N(S; (1=2)n) log(3n) n log 3 log 3 = lim = lim = lim = ; n!1 log(1=(1=2)n) n!1 log(2n) n!1 n log 2 log 2 using (4.4). Let us look at S from another point of view. Letting fc1; c2; c3g be the n n vertices of S0, we de ne the contractions Ti: R ! R by 1 T (x) = (x c ) + c ; i = 1; 2; 3: i 2 i i 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 log 3 dim S = s = : S log 2 Thus with reference to Proposition 5.11 we have log 3 dim S = dim S = dim S = dim S = : H P M S log 2 Without thorough calculations, we also have the following from Proposition 4.16: log 3 dim (S ¢ I) = dim S + dim I = + 1: H H H log 2 since dimH I = dimM I = 1: Proposition 7.1. The topological dimension of the Sierpinski gasket is

dimT S = 1: 35

Proof. To narrow it down we use Proposition 5.1 and conclude that dimT S  1 since dimH S = 1:584 :::. We also know that S 6= ; so dimT S  0. Now, since S is connected in its perimeter we have that dimT S 6= 0 by Proposition 2.14 and thus dimT S = 1. Let us now sum this up with a proposition. Proposition 7.2. If S is the Sierpinski gasket then log 3 (S) = dim S dim S = 1 = 0:584 :::; H T log 2 and thus S is a fractal set.

Proof. The proposition follows from Proposition 5.3 since dimH S 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 dimM E. 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 + 21; i.e. three squares to cover S and two squares to cover the line segment. N(E; 1=4) = 13 = 9 + 4 = 32 + 22; and in general: N(E; (1=2)n) = 3n + 2n and thus log N(E; (1=2)n) dimM E = lim n!1 log(1=(1=2)n) log(3n + 2n) = lim n!1 log(2n) log(3n(1 + ( 2 )n) = lim 3 n!1 log(2n) log(3n) + log(1 + ( 2 )n) = lim 3 n!1 log(2n) log(3n) = lim n!1 log(2n) log 3 = log 2 log 3 = maxf ; 1g log 2

= maxfdimM S; dimM Ig; 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 di erent 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 k this we have the rst 2 colored rows correspond to Sk. 38 Chapter 7. The Sierpinski gasket 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.

Leifsson, 2006. 39 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 2 us K2 consisting of 4 = 16 line segments of equal length. Thus K3 consists 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 Kn: n!1 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.

2 2 Proof. Let K0 have length 1. Then K1 has length 4=3, K2 has length 4 =3 and n n so on. Thus Kn constructed after n steps is of length 4 =3 . Hence intuitively K should have length 4n lim = 1: n!1 3n For a rigorous proof, we take an advance look at Proposition 8.3 below and notice that dimH K > 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: p 1 3i T = ( + )z 1 2 6 and p 1 3i T = ( )(z 1) + 1: 2 2 6

Thus K is an invariant set under these contractions. Wep can nowp conclude that 1 3i the Moran equation (5.2) is satis ed with c = j 2 + 6 j = 1= 3 and m = 2, giving us log 4 dim K = ; S 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 log 4 dim K = dim K = dim K = dim K = : H P M S log 3 Proof. The proposition follows from our recent arguing above and from Propo- sition 5.11.

Let us con rm dimM K = dimS K with an example. 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, 2 N(K;"2) = N(K; 1=9) = N(K; 1=3 ) = 12 = 3 ¡ 4, 3 2 N(K;"3) = N(K; 1=27) = N(K; 1=3 ) = 48 = 3 ¡ 4 , n n 1 and in general N(K;"n) = N(K; 1=3 ) = 3 ¡ 4 . Hence

log N(K;"n) log N(K;"n) dimM K = lim = lim "n!0 log(1="n) n!1 log(1="n) log N(K; 1=3n) log(3 ¡ 4n 1) = lim = lim n!1 log(1=(1=3)n) n!1 log(3n) (n 1) log 4 + log 3 log 4 = lim = : n!1 n log 3 log 3

Proposition 8.5. If K is the von Koch curve then dimT K = 1.

Proof. First of all, since dimH K = 1:261 ::: we know that dimT K  1 by Proposition 5.1. And since K is not the empty set we can also conclude that dimT K  0. From Proposition 2.14 we have that dimT K 6= 0 because K is connected. Proposition 8.6. The von Koch curve K is a fractal set with the fractal degree log 4 (K) = 1: log 3 42 Chapter 8. The von Koch snow ake

Proof. From Proposition 5.3 we have that K is a fractal since dimH K is non- integer and remembering De nition 5.2 we have that the fractal degree of K is log 4 (K) = dim K dim K = 1 = 0:2618 :::: H T log 3

8.1 The von Koch curve versus C2

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

log 2 dim (Cn) = dim (Cn) = dim (Cn) = dim (Cn) = n dim C = n H P M S H log 3 (8.1) where Cn = C ¢ C ¢ ¡ ¡ ¡ ¢ C  Rn: | {z } n times Now, from (8.1) we have that

log 2 dim C2 = 2 ; H log 3 which leads us to the rst interesting observation that

2 dimH C = dimH K:

Also, noticing that C2 is totally disconnected whilst K is not we know that their topological dimensions di er, and thus their fractal degrees do as well. 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, n f£ = f([0; 1]), is strictly positive, i.e. L (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 P1 of 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. 2 To generate P2 we now think of the unit square as 16 = 4 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=2n 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 P5 follows below. The length of Pn is obviously 1 2n ; 2n

Leifsson, 2006. 43 44 Chapter 9. Space- lling curves

Figure 9.1: The rst iteration, P1, of the Hilbert curve. thus P = lim Pn n!1 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 p 2 jP (t) P (t)j  for t 2 [0; 1]: n+1 n 2n Thus for m > n we have that mX 1 sup jPm(t) Pn(t)j  sup jPi+1(t) Pi(t)j 0t1 0t1 p p i=n X1 2 2  = 2 ¡ ! 0; as m; n ! 1: 2i 2n i=n 1 Thus (Pn)n=1 is uniformly convergent according to the Cauchy criterion for 1 uniform convergence, which states that a sequence, (Pn)n=1, of functions de ned on a set E  Rn is uniformly convergent if and only if

sup jPm(t) Pn(t)j ! 0 as m; n ! 1: t2E 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 L2([0; 1]2) = 1 > 0 the claim thus follows. From (5.3), with m = 4 and c = 1=2, we get log 4 dim P = = 2: S log 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 dimH P = 2 and so

dimH P = dimP P = dimM P = dimS P = 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 ) = dimH P dimT P = 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 di erent ways. We will use line segments to construct it. Our starting set, D0, consists of the unit interval.p To create D1 we replace D0 with two line segments each of length d1 = 1= 2 joined at a right angle. Next we receive D2 by replacing each line segment in D1 with two line segments each of length 1 1 d = p d = ; 2 2 1 2 46 Chapter 9. Space- lling curves again joined at a right angle. If we let this procedure continue ad in mum we in each iteration Dn, n = 0; 1; 2;:::, have that every line segment is of length 1 d = (p )n n 2 and the Heighway dragon, D, is the limit set received, i.e.

D = lim Dn: n!1 A simple illustration of the rst ve iterations is provided below, starting with D0 in the upper left corner.

Figure 9.3: The rst ve iterations of the Heighway dragon curve.

Proposition 9.4. None of the approximations Dn, n = 0; 1; 2;:::, of the Heigh- way dragon curve crosses itself. Proof. See e.g. [7] p. 21. An intuitive understanding of the validity of Proposition 9.4 can be gained by rounding the right angles o a bit. An example where this has been done to D9 is depicted below. It can also be shown in a similar manner as in the case of the Peano curve that the Heighway dragon is a space- lling curve, (see Proposition 2.4.3 in [7]).

Proposition 9.5. If D is the Heighway dragon curve we have that

dimH D = dimP D = dimM D = dimS D = 2: p Proof. The reduction ratio in every step is c = 1= 2 and in each step every line segment is replaced by m = 2 new ones. Thus by (5.3) we have log 2 dim D = p = 2: S log 2

Remarks 3.8 and 3.10 imply dimH D = 2 and thus the claim follows thereafter from Proposition 4.14 and Remark 5.12. 9.2. The Heighway dragon 47

Figure 9.4: The D9 approximation of the Heighway dragon with the right angles rounded o .

One can easily create di erent variants of the Heighway dragon by changing the angles. One among many interesting manipulations of the Heighway dragon leads to an approximation of the Sierpinski gasket. In the following example we will change the angles in the Heighway dragon curve which will give us an altered Heighway dragon curve, D0. Now, joining three of these in an equilateral triangle results in a set known as the fudge ake (from fudging the von Koch snow ake), which is illustrated below.

Figure 9.5: The Fudge ake, consisting of three copies of D0 put together.

Example 9.6. Changing the angles in the dragon curve from 90 degrees to 120 0 and using the same starting set, i.e. the unit line, gives us that D1 has side lengths 1 sin 60 = p1 . Each of the 2n line segments in the n:th step has length 2 3 ( p1 )n and thus intuitively D0's total length is in nite since 3

 2 n lim p = 1: n!1 3 48 Chapter 9. Space- lling curves

However, log 2 log 2 dim D0 = dim D0 = p = 2 > 1; S H log 3 log 3 thus by Remark 3.10 it has in nite length. A noticeable fact here is also that 0 dimS D is twice as big as dimS C and equal to dimS K. Chapter 10

Conclusions and nal remarks

Although this thesis is restricted to some basic parts of fractal analysis I think it is enough to emphasize the need of fractal analysis as an important complement when our classical calculus does not apply. The thesis could easily have been broadened by e.g. including other metric spaces besides Rn. However, to allow more readers with the adequate prerequisites and to get a more connected text, this has been avoided. There are also many other dimension concepts and measures not mentioned which could have been included. To sum it up, I think the purpose of collecting similar concepts has succeeded and connections between dimensions and certain fractals have been clari ed. It would have been interesting to extend the thesis with some areas applicable to fractal analysis such as number theory, however as mentioned in the introduction the area of fractal analysis is wide but still much is unexplored within it. Overall, I think fractal analysis is an enjoyable and rewarding area to study. Even though the theory in the rst chapters is occasionally quite heavy it is interesting to see when the di erent dimensions coincide and under which con- ditions. Then applying the theory to various fractal sets and to investigate how they are connected and di er makes it well worthwhile I think.

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Upphovsratt Detta dokument hallstillgangligtpaInternet - eller dess framtida ersattare - under 25 arfranpubliceringsdatum under forutsattningatt inga extraordi- naraomstandigheteruppstar. Tillgangtill dokumentet innebartillstandfor var och en att lasa,ladda ner, skriva ut enstaka kopior forenskilt bruk och att anvandadet oforandratforickekommersiell forskning och forundervisning. Overforingav upphovsrattenvid en senare tidpunkt kan inte upphava detta tillstand.All annan anvandningav dokumentet kraver upphovsmannens med- givande. Foratt garantera aktheten,sakerheten och tillgangligheten nns det losningarav teknisk och administrativ art. Upphovsmannens ideella rattin- nefattar rattatt bli namndsom upphovsman i den omfattning som god sed kraver vid anvandningav dokumentet paovan beskrivna sattsamt skydd mot att dokumentet andras eller presenteras i sadanform eller i sadant sammanhang som arkrankande forupphovsmannens litteraraeller konstnarligaanseende eller egenart. Forytterligare information om LinkopingUniversity Electronic Press se forlagetshemsida http://www.ep.liu.se/ c 2006, Patrik Leifsson

Leifsson, 2006. 53