# Examensarbete Fractal sets and dimensions Patrik Leifsson

Examensarbete Fractal sets and dimensions Patrik Leifsson LiTH - MAT - EX - - 06 / 06 - - SE 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 Sprak Datum Division, Department Date Matematiska Institutionen 581 83 LINKOPING SWEDEN May 2006 Rapporttyp Report category Language x Avdelning, Institution Svenska/Swedish Engelska/English x Licentiatavhandling Examensarbete ISBN ISRN LiTH - MAT - EX - - 06 / 06 - - SE C-uppsats Serietitel och serienummer D-uppsats Title of series, numbering ISSN 0348-2960 Ovrig rapport URL for elektronisk version http://www.ep.liu.se/exjobb/mai/2006/tm/006/ Titel Fractal sets and dimensions Forfattare Patrik Leifsson Title Author Sammanfattning Abstract Nyckelord Keyword 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 denitions of fractal dimensions, which can be used to measure a set's fractal degree. Comparisons of these denitions 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 denitions 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. 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 introduce certain denitions of fractal dimensions, which can be used to measure a set's fractal degree. Comparisons of these denitions 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 denitions referring to similar concepts. Keywords: box dimension, Cantor dust, Cantor set, dimension, fractal, Haus- dor dimension, measure, Minkowski dimension, packing dimension, Sierpinski 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 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. Leifsson, 2006. ix x Preliminaries In this section we collect some basic notations and denitions. x y means that there exists c > 0 such that xc < 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 e0 2 Rn and R 3 " > 0, we dene an open ball, B , as B (e0 ; ") = fe 2 Rn : je e0 j < "g; and a closed ball, B , as B (e0 ; ") = fe 2 Rn : je e0 j "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 bounded, i.e. Rn is bounded if the diameter of E , diam E , is diam E = supfjx yj : x; y 2 E g < 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 x0 2 Rn 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 dened 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 F: Contents 1 Introduction 1.1 Purpose of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Structure of the thesis . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 The topological dimension 2.1 2.2 2.3 2.4 The small inductive dimension The large inductive dimension . The covering dimension . . . . The topological dimension . . . . . . . . . . . . . . . . . . . 3 The Hausdor measure and dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 The Hausdor measure . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The Hausdor dimension . . . . . . . . . . . . . . . . . . . . . . 1 1 1 2 3 3 3 4 4 7 7 9 4 Minkowski dimensions 13 5 Fractals and self-similarity 23 6 Cantor sets 27 7 The Sierpinski gasket 33 8 The von Koch snowake 39 9 Space-lling curves 43 10 Conclusions and nal remarks 49 4.1 The packing dimension . . . . . . . . . . . . . . . . . . . . . . . . 19 4.2 Product relations . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 5.1 Fractals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 5.2 Self-similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 6.1 The ternary Cantor set . . . . . . . . . . . . . . . . . . . . . . . . 27 6.2 Cantor set using ternary numbers . . . . . . . . . . . . . . . . . . 28 7.1 The Sierpinski gasket using the ternary tree . . . . . . . . . . . . 36 7.2 The Sierpinski sieve . . . . . . . . . . . . . . . . . . . . . . . . . 36 8.1 The von Koch curve versus C 2 . . . . . . . . . . . . . . . . . . . 42 9.1 Peano space-lling curve . . . . . . . . . . . . . . . . . . . . . . . 43 9.2 The Heighway dragon . . . . . . . . . . . . . . . . . . . . . . . . 45 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 denition of what should be considered fractals. In this denition 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 denition 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 1 X 3 f (x) = ai cos(bi x); 0 < a < 1; ab > 1 + ; 2 i=0 which is continuous everywhere but nowhere dierentiable. Another classical example from this period is the triadic Cantor set, which will be studied thoroughly later on in this thesis. The use of fractal analysis is wide. It ranges from probability theory, physical 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 fullled 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 Leifsson, 2006. 1 2 Chapter 1. Introduction which are later on put to the test. In Chapter 5 we look at Mandelbrot's denition 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 satised, concerning the dimension dim A of A: 8 < (i) dimfag = 0; where fag is the singleton set. (ii) dim I 1 = 1; where I 1 is the unit interval. (I) : (iii) dim I m = m; where I m is the m-dimensional hypercube. (II) Monotonicity: If A E then dim A dim E . n (III) Countable stability: If fAi g1 i=1 is a sequence of closed subsets of R , then 1 [ dim Ai = sup dim Ai : i 1 i=1 (III') Finite stability: If A1 ; A2 ; : : : ; Am are closed subsets of Rn , then dim m [ 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. L1 jx yj jg(x) g(y)j L2 jx 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 fullls condition (IV') is also known as a lipeomorphism. Chapter 2 The topological dimension There are three dierent denitions of the topological dimension: ind, Ind and Cov. The rst two are inductively dened. 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 Denition 2.1. The small inductive dimension of a set E inductively as follows: Rn is dened 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. Denition 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 ). Denition 2.3. The large inductive dimension of a set E inductively as follows: Rn is dened 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 dene the covering dimension. Denition 2.4. A family F of subsets of S E F. Rn is a cover of a set E Rn if F 2F Denition 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 renement of E . Denition 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. Denition 2.8. The covering dimension of a set E Rn is dened 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 renement 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 Denition 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 B g 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-Bernstein theorem [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 simplied 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 dene the topological dimension of E as dimT E = ind E: Denition 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 , fullls the dimension requirements 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 Ix;" = (x "; x + "); x 2 I 1 = 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. (I)(iii) That dimT I m = m can be shown by an argument similar to that for (I)(ii). (II) Let A E Rn 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 important. 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 Denition 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 but dimT I = dimT Q = 0; 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. Proposition 2.19. (Theorem A.4.14 in [6]) dimT Rn = 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 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 1 S Denition 3.1. We say that fEi g1 Ei and i=1 is a -cover of a set E if E i=1 0 diam Ei , for all i. For a set E Rn , s 0, and > 0, dene Hs (E ) = inf 1 X i=1 (diam Ei )s ; where the inmum are taken over all (countable) -covers fEi g1 i=1 of E . Also dene the s-dimensional Hausdor measure by Hs (E ) = lim Hs (E ) = sup Hs (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 denition. Denition 3.2. An outer measure, , on subsets of Rn that satises: Rn is a positive set function on all (;) = 0; monotonicity: (A) (E ) if A E Rn ; countable sub-additivity: for all Ei Rn . 1 [ i=1 Ei 1 X i=1 (Ei ) Proposition 3.3. Hs is an outer measure. Leifsson, 2006. 7 8 Chapter 3. The Hausdor measure and dimension Proof. With the above denition of outer measure we have Hs (;) = 0 since diam ; = 0 . Let A E Rn and " > 0. that 1 X i=1 Then there is a -cover fEi g1 1=1 of E such (diam Ei )s Hs (E ) + ": But fEi g1 1=1 is also a valid -cover of A and thus 1 X s H (A) (diam Ei )s Hs (E ) + ": i=1 Now, letting " ! 0 gives us Hs (A) Hs (E ): 1 s P H (Ei ) < 1 we have for an arbitrary " > 0 that for each i=1 i 1 there exists a -cover fAij g1 j =1 of Ei such that Assuming 1 X j =1 (diam Aij )s < Hs (Ei ) + 2 i ": Thus fAij gi;j 1 constitutes a valid -cover of X (diam Aij )s i;j 1 1 S i=1 Ei and 1 1 X X Hs (Ei ) + 2 i " = Hs (Ei ) + ": i=1 i=1 Letting " ! 0 proves the claim. Proposition 3.4. Hs is a measure, i.e. if fAi g1 i=1 is a pairwise disjoint countable collection of measurable sets, then 1 X 1 [ Ai = (Ai ): i=1 i=1 (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 = fqi g [0; 1], for each qi in Q \ [0; 1], we have 1 X H1 (Ai ) = inf (diam Eij )1 = 1; (3.2) j =1 3.2. The Hausdor dimension 9 where Eij are -covers of Ai with diam Eij sian products fqi g [0; 1] we have . Since we are considering Carte- 1 X (diam Eij ) 1 j =1 and thus (3.2) is fullled. Then using balls, Bi , of radii to cover E we need approximately c 12 of these, where c 2 R. Thus H 1 1 [ i=1 Ai = inf 1 X i=1 (diam Bi ) c 1 2 1 < 1 X i=1 H1 (Ai ) = 1: Hence the necessary condition (3.1) is not fullled in order for measure. Hs to be a Denition 3.7. The Lebesgue measure Ln on Rn is dened as follows. For A of the form dene A = f(x1 ; : : : ; xn ) 2 Rn : ai xi bi g (3.3) Ln (A) = (b1 a1 )(b2 a2 ) (bn an ); and extend Ln to general subsets of Rn by 1 1 o nX [ Ln (Ai ) : A Ai for all Ai of the form (3:3): Ln (A) = inf 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 dene the Hausdor dimension. Denition 3.9. The Hausdor dimension of a set E Rn is dimH (A) = supfs : Hs (E ) > 0g = supfs : Hs (E ) = 1g = inf ft : Ht (E ) < 1g = inf ft : Ht (E ) = 0g: Remark 3.10. The previous denition can also be expressed as if s< dimH (E ); Hs (E ) = 1 0 if s> dim (E ); H and Hs (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 Hausdor measure of the set E so to speak jumps from innity to zero. Proposition 3.11. The Hausdor dimension, dimH , satises 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 dimH fag = inf fs : Hs (fag) = 0g = 0: (I)(ii) We have 0 < H1 (I 1 ) < 1 and hence dimH I 1 = 1, using Remark 3.10. (I)(iii) Let I m be an m-dimensional hypercube in Rn , 1 m n. We have Hm (I m ) = cLm (I m ) 2 (0; 1): Hence dimH I m = m, using Remark 3.10. (II) Monotonicity: Let A E Rn . Since Hs (A) Hs (E ) when A E Rn , it follows from Denition 3.9 that dimH (A) dimH (E ). (III) Countable stability: From the monotonicity we have for all j , dimH 1 [ i=1 Ei dimH Ej and hence, taking supremum over all j , dimH 1 [ The inequality dimH j =1 1 [ i=1 Ej Ei sup dimH Ej : j 1 sup dimH Ej j 1 follows from the fact that if s > dimH Ei for all i, then Hs (Ei ) = 0 S 1 for all i, which gives us Hs Ei = 0 and hence i=1 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 4 has Hausdor dimension log is homeomorphic to [0,1] with Hausdor dimension log 3 1 and thus we can directly see that invariance is not fullled 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 justied by the fact that for a singleton set Ei we have H0 (Ei ) = 1: Thus dimH (Ei ) = 0: Hence, by the countable stability dimH [1 E = 0: i=1 i 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 dimension, 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 dened through an upper and lower dimension, which need not coincide. Denition 4.1. For a non-empty bounded subset E of Rn we dene the upper Minkowski dimension as dimM E = inf fs : lim sup N (E; ")"s = 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 dene the lower Minkowski dimension as dimM E = inf fs : lim inf N (E; ")"s = 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 E g: (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. S Now, if there exists an x in E n ki=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); 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 Denition 4.1 and Proposition 4.2 that dimM E = lim sup "!0+ log P (E; ") log N (E; ") = lim sup log(1=") "!0+ log(1=") (4.2) dimM E = lim inf log P (E; ") log N (E; ") = lim inf : "!0+ log(1=") log(1=") (4.3) and "!0+ 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 N (E; ")"s < ; for all " < "0 : Then so log N (E; ")"s < log s> log log N (E; ") ; log " and letting " ! 0+ and taking inmum over all s > dimM E , we get dimM E lim sup "!0+ log N (E; ") : log(1=") Now suppose log N (E; ") < s < dimM E: log(1=") Then there exist "j ! 0 and 0 > 0 such that lim sup "!0+ N (E; "j )"sj > 0 > 0 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 ) N (E; ") lim sup loglog(1 ; =") "!0+ which is a contradiction. Thus dimM E = 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 log P (E; ") log N (E; ") = lim : "!0 log(1=") "!0 log(1=") dimM E = lim (4.4) In this denition of the Minkowski dimension, the number N (E; ") can be replaced 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 dened by means of the n-dimensional volume of an "-neighbourhood of E Rn . The "-neighbourhood, E" , of E is dened as E" = fx 2 Rn : jx yj " for some y 2 E g: (4.5) Then, using the above dened Lebesgue measure, we have the following equivalent formulas for the Minkowski dimension. Proposition 4.3. Let E Rn . Then dimM E = n + lim sup "!0+ log Ln (E" ) ; log(1=") (4.6) dimM E = n + lim inf log Ln (E" ) ; log(1=") (4.7) "!0+ and if it exists. log Ln (E" ) "!0+ log(1=") dimM E = n + lim (4.8) 16 Chapter 4. Minkowski dimensions Proof. First we prove that the inequalities c"n P (E; ") Ln (E" ) c(2")n 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 Ln (E" ) c(2")n N (E; "): To understand that c"n P (E; ") Ln (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 log(c 1 " n Ln (E" )) log P (E; ") lim sup log(1=") "!0+ "!0+ log(1=") log Ln (E" ) log c n log " = lim sup log " "!0+ log Ln (E" ) = lim sup +n log " "!0+ log Ln (E" ) = n lim inf : log " "!0+ dimM E = lim sup This was attained from c"n P (E; ") Ln (E" ). Using the last inequality in (4.9) we have log N (E; ") log(c 1 (2") n Ln (E" )) lim sup log(1=") "!0+ log(1=") "!0 + n log L (E" ) log c n log(2") = lim sup log " "!0+ n log L (E" ) = lim sup +n log " "!0+ log Ln (E" ) = n lim inf : log " "!0+ dimM E = lim sup Thus we conclude that dimM E = n lim inf + "!0 log Ln (E" ) ; log " which proves (4.6). Now, (4.7) follows in an manner analogous to the above. Finally, log Ln (E" ) dimM E = n + lim+ "!0 log(1=") follows from (4.6) and (4.7). 17 From the denition of the Hausdor measure we can deduce the useful relation Hs (E ) N (E; ) s . In other words, if fAi g1 i=1 is a -cover of E , then as diam Ai for all i, we have 1 X i=1 s s (diam Ai )s |s + s + {z + } = N (E; ) : N (E;) Theorem 4.4. For all E Rn , dimH E dimM E dimM E: (4.10) Proof. If s < dimH E , then 0 < Hs (E ) = lim Hs (E ) ! lim N (E; ) s !0+ 0+ and thus log N (E; ) + s log > log Hs (E ) 1, for small enough this we have s lim inf !0+ > 0. From log N (E; ) log Hs (E ) = dimM E dimM E: 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 to its Minkowski dimension. The set E = f0; 1; 21 ; 13 ; 14 ; : : :g has Hausdor dimension 0 since it is countable. However, dimM E = 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 " k12 . The number of balls of radius " it takes to cover the points 1; 12 ; 13 ; 14 ; : : : ; k 1 1 equals k 1 p1" . And to cover the points which lie in E \ [0; k1 ] by balls of radius ", it takes about 1 2p1 " balls. Hence the number of balls needed to cover E is essentially 2k" N (E; ") 1 1 1 p +p p 2 " " " from which we obtain dimM E = log N (E; ") = "!0+ log " lim 1 log( p1" ) log " 1 = lim 2 = : "!0+ log " "!0+ log " 2 lim A question that still remains is whether the Minkowski dimension fullls the dimension requirements from Section 1.3. Proposition 4.6. The upper Minkowski dimension, dimM , satises conditions (I), (II), (III') and (IV') from Section 1.3. The lower Minkowski dimension, dimM E , fullls conditions (I), (II) and (IV') of these. 18 Chapter 4. Minkowski dimensions Proof. (I)(i) We have dimM fag = inf fs : lim sup N (fag; ")"s = 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 (I m ; ") " m and hence log N (I m ; ") = m: "!0+ log(1=") dimM I m = dimM I m = dimM I m = lim (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). (III') Let fEij g1 j =1 be a -cover of Ai for i = 1; : : : ; n. Then Sn fEij ; i = 1; : : : ; n; j = 1; : : : ; 1g is a -cover of i=1 Ai , so N n [ i=1 Ai ; n X i=1 N (Ai ; ): If s > dimM Ai for all i, then we have in accordance with Denition 4.1 that lim sup N (Ai ; ) s = 0: !0+ Adding over i = 1; : : : ; n gives us lim sup N !0+ n [ i=1 Ai ; s = 0: S Thus from Denition 4.1 we have dimM ( ni=1 Ai ) s. Hence dimM ( n [ i=1 Ai ) i=1 max dimM Ai : ;:::;n S Now, suppose that dimM ( ni=1 Ai ) < max dimM Ai , i.e. there is an i such i=1;:::;n that n [ dimM Ai > dimM Ai : i=1 4.1. The packing dimension 19 But this contradicts the monotonicity. Thus dimM n [ Ai = i=1 max dimM Ai : ;:::;n i=1 (IV') The Lipschitz invariance is fullled 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 opposite inequality, ([9], p. 44). 1 instead, gives us the 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; 21 ; 14 ; 18 ; : : :g. Since F E , where E is the set considered in Example 4.5, we immediately know that dimM F dimM E = 21 from the monotonicity condition. We also know that dimH F = 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 log N (F; 2 n ) n!1 log 2n lim we have that log n log n nlim = lim = 0; n !1 log 2 n!1 n log 2 dimM F = dimM F = 0: Now consider the mapping :x! 1 x1 2 1 ; from E to F for each nonzero element x in E . Since 1 x1 1 = 0; lim 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 fullled for dimM . However, these complications can be overcome if we introduce the packing dimension for a set E Rn . 20 Chapter 4. Minkowski dimensions Denition 4.9. The lower and upper packing dimensions are n o dimP E = inf sup dimM Ei and i1 n o dimP E = inf sup dimM Ei ; i1 where the inma are taken over all countable covers fEi g1 i=1 of E bounded sets. Rn by With this denition, we get dimP E = dimP E = 0 when E is countable, and the countable stability criterion for the packing dimension is fullled. The other dimension requirements valid for the Minkowski dimension are still valid. Denition 4.10. Let 0 < " < 1 and 0 s < 1. For a set E Rn we dene P"s (E ) = sup 1 X i=1 (diam Bi )s and P s (E ) = lim P"s (E ) = inf P"s (E ): "!0+ ">0 The supremum is taken over all collections of disjoint balls fBi g of radius less or equal to " and centers in E . We then dene the s-dimensional packing measure of a set E Rn as 1 1 o nX [ s s P (Ei ) : E = Ei : P (E ) = inf i=1 Proposition 4.11. i=1 P s (E ) is a measure on Rn . Proof. See for example [16] p. 82. We now use the s-dimensional packing measure to dene the packing dimension. Theorem 4.12. For a set E Rn , dimP E = inf fs : P s (E ) = 0g = inf fs : P s (E ) < 1g = supfs : P s (E ) > 0g = supfs : P s (E ) = 1g: Proof. See Theorem 5.11 in [16]. Denition 4.13. For E Rn , we dene 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 ) P s (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 Denition 3.9 and Theorem 4.12. The following proof of the last inequality 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 ) P s (E ) = 1: So for 0 < 1 there are disjoint balls fBi g1 i=1 with centers in E and radii at most equal to such that 1 X 1 < (diam Bi )s : i=1 Next we assume that for all k we have that nk of these balls satisfy 2 k 1 < diam Bi 2 k : Then 1< 1 X k=0 nk 2 ks (4.11) is also satised. 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 ) nk and (2 k 1 )t P (E; 2 k 1 ) nk (2 k 1 )t > 2 t (1 where 2 k 1 2t s ) ; < . Thus lim sup P (E; ) t 2 t (1 !0 2t s ) > 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 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 ) dimH A + dimH E ; dimH (A E ) dimH A + dimM E ; dimM (A E ) dimM A + dimM E ; (4.12) (4.13) (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 disconnected set E Rn 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]). Denition 5.2. We say that a set E Rn 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. Denition 5.4. For a closed set E Rn , the mapping T : E contraction on E if there is a c 2 (0; 1) such that jT (x) T (y)j cjx yj ! E is called a (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. Denition 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 dened 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 fullling (5.1) we call T a similar contraction. Denition 5.7. For a family T = fT1 ; T2 ; : : : ; Tm g of similarities with contraction 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 Denition 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. Denition 5.9. We say that the contractions T1 ; T2 ; : : : ; Tm fulll 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. Denition 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 cs1 + cs2 + + csm = 1: (5.2) In the special case when c1 = c2 = = cm = c we have that Thus mcs = 1 and hence log m + s log c = 0: log m : (5.3) 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. dimS E = s = 5.2. Self-similarity 25 Proposition 5.11. (Theorem 2.7 in [10], Theorem 4.14 in [16]) Let Ti be similarities on Rn satisfying the open set condition with contraction ratios ci , i = 1; 2; : : : ; m. If E is the invariant set of fTi gm i=1 , then dimH E = dimP E = dimM E = dimS E ; 0 < Hs (E ) < 1 and P s (E ) < 1, where s = dimS E ; There exist e1 ; e2 2 (0; 1) such that for s = dimS E , e1 rs Hs (E \ B (x; r)) e2 rs for all x 2 E and 0 < r 1. Remark 5.12. If the open set condition is not fullled 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 with contraction ratios ci < 1; i = 1; : : : ; m, on a closed invariant set E Rn we have that where (5.2) is fullled. dimH E s and dimM E s; 26 Chapter 5. Fractals and self-similarity Chapter 6 Cantor sets Generally, for 0 < < 12 , we dene Cantor sets on R as the limit set C () = 1 [ 2i \ i=0 j =1 Ei;j where E0;1 = [0; 1]; E1;1 = [0; ]; E1;2 = [1 ; 1], and for dened intervals Ei 1;1 ; : : : ; Ei 1;2i 1 , the intervals Ei;1 ; : : : ; Ei;2i are dened through removing intervals of length (1 2) diam Ei 1;j = (1 2)i 1 from the middle of each interval Ei 1;j . Thus each Ei;j has length i . The following proposition will be veried 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 =) . Hs (C ()) = 1, where s = dimH C (). 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 [ 2i \ i=0 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 C1 we remove the open middle third interval from C0 , i.e. C1 = C0 n ( 13 ; 23 ) = [0; 13 ] [ [ 32 ; 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 1 1 2 1 1 1 2 = : 3 3 9 Thus C2 = C1 n (( 91 ; 92 ) [ ( 79 ; 89 )) = [0; 19 ] [ [ 29 ; 31 ] [ [ 23 ; 79 ] [ [ 89 ; 1]: Continuing like this in innitely 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 generations 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 + 21 + 20 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 1 1 1 1 = + + + = 0:1011012 : 64 2 8 16 64 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 = 0:1000:::3 = 0:0222:::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. Proof. Consider the second generation C1 of C and the binary sequences (xi )1 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 sequences (xi )1 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 Tset is compact since it is both closed and bounded. It is closed since in C = 1 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 = 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 ; c2 2 I gives us that a+b a+b c1 < < c2 and 2= C: 2 2 Thus there is a c = a+2 b 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 T1 = i.e. x 3 and T2 = 1 x C = T1 (C ) [ T2 (C ): 3 ; 30 Chapter 6. Cantor sets From (5.3) we thus have log 2 : log 3 The open set condition is fullled with O = (0; 1). Hence we have dimS C = dimH C = dimP C = dimM C = dimS C = 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 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 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 log N (Cn ; (1=3)n ) 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 dimM C = lim Proposition 6.5. If C is the ternary Cantor set we have (C ) = dimH C dimT C = log 2 >0 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 C 02 . From Proposition 6.1 we have that an thus dimH C 0 = log 2 1 = ; log 4 2 dimH C 02 = 2 1 2 6.2. Cantor set using ternary numbers 31 by Proposition 4.16. Now, C 02 is obviously totally disconnected and thus dimT C 02 = 0, which also can be seen by using Proposition 4.17 which gives us dimT C 02 dimT C 0 + dimT C 0 = 0; since C 0 is totally disconnected as well. And since C 02 is not the empty set we know that dimT C 02 0. Hence the fractal degree of C 02 is (C 02 ) = dimH C 02 dimT C 02 = 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 innitely 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 41 . 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 Thus Leifsson, 2006. N (S; "n ) = N (S; (1=2)n ) = 3n : log N (S; "n ) log N (S; "n ) = lim n!1 log(1="n ) "n !0+ log(1="n ) dimM S = lim 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 = ; 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 ; c3 g be the vertices of S0 , we dene the contractions Ti : Rn ! Rn by 1 Ti (x) = (x ci ) + ci ; i = 1; 2; 3: 2 Thus S = T1 (S ) [ T2 (S ) [ T3 (S ); with the contraction ratio 1/2 for each Ti . Now, the Sierpinski gasket satises the open set condition if we let O be the interior of the starting triangle, S0 . Hence, by (5.3), we have log 3 dimS S = s = : log 2 Thus with reference to Proposition 5.11 we have = lim dimH S = dimP S = dimM S = dimS S = log 3 : log 2 Without thorough calculations, we also have the following from Proposition 4.16: dimH (S I ) = dimH S + dimH I = since log 3 + 1: log 2 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 (S ) = dimH S dimT S = log 3 log 2 1 = 0:584 : : : ; 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: and thus N (E; (1=2)n ) = 3n + 2n log N (E; (1=2)n ) n!1 log(1=(1=2)n ) log(3n + 2n ) = lim n!1 log(2n ) log(3n (1 + ( 23 )n ) = lim n!1 log(2n ) log(3n ) + log(1 + ( 23 )n ) = lim n!1 log(2n ) n log(3 ) = lim n!1 log(2n ) log 3 = log 2 log 3 = maxf ; 1g log 2 = maxfdimM S; dimM I g; dimM E = lim 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 indenitely, 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 Sk . 38 Chapter 7. The Sierpinski gasket Chapter 8 The von Koch snowake We will now consider another fractal set called the von Koch snowake, illustrated below. Figure 8.1: The von Koch snowake. The von Koch snowake 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 snowake. Leifsson, 2006. 39 40 Chapter 8. The von Koch snowake 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 K3 consists of 43 = 64 line segments and so on. K is now dened as the limit set obtained after innitely many iterations, i.e. K = nlim !1 Kn : We now look at one of the properties of K and hence the von Koch snowake thereby. Proposition 8.1. The von Koch curve is of innite length. Proof. Let K0 have length 1. Then K1 has length 4=3, K2 has length 42 =32 and so on. Thus Kn constructed after n steps is of length 4n =3n . Hence intuitively K should have length 4n lim n = 1: n!1 3 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 innite. Corollary 8.2. The von Koch snowake has innite length. The von Koch curve is obviously self-similar, while the von Koch snowake is not since it has no copies of itself in its structure. The open set condition is fullled with O being the open equilateral triangle with side length equal to K0 . K can be dened through the following contractions using complex numbers: p 1 3i T1 = ( + )z 2 6 and p 3i 1 )(z 1) + 1: T2 = ( 2 6 Thus K is an invariant set under these contractions. We p can now p conclude that the Moran equation (5.2) is satised with c = j 12 + 63i j = 1= 3 and m = 2, giving us log 4 dimS K = ; 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 dimH K = dimP K = dimM K = dimS K = log 4 : log 3 Proof. The proposition follows from our recent arguing above and from Proposition 5.11. Let us conrm 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, 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 log N (K; "n ) log N (K; "n ) dimM K = lim = lim n!1 log(1="n ) "n !0 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 (K ) = log 4 log 3 1: 42 Chapter 8. The von Koch snowake Proof. From Proposition 5.3 we have that K is a fractal since dimH K is noninteger and remembering Denition 5.2 we have that the fractal degree of K is log 4 (K ) = dimH K dimT K = 1 = 0:2618 : : : : log 3 8.1 The von Koch curve versus C 2 We now briey compare the von Koch curve, K , with the Cantor product C 2 = C C , where C is the middle third Cantor set. Figure 8.3: The Cantor product C 2 . With the dimension formulas in Section (4.2) we directly have dimH (C n ) = dimP (C n ) = dimM (C n ) = dimS (C n ) = n dimH C = n where log 2 log 3 (8.1) n Cn = C | C {z C} R : n times Now, from (8.1) we have that dimH C 2 = 2 log 2 ; log 3 which leads us to the rst interesting observation that dimH C 2 = dimH K: Also, noticing that C 2 is totally disconnected whilst K is not we know that their topological dimensions dier, and thus their fractal degrees do as well. Chapter 9 Space-lling curves Before digging into any examples of space-lling curves we shall dene its concept. Denition 9.1. A continuous function f : [0; 1] ! Rn , n 2, is called a spacelling 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 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. 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=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 2n Leifsson, 2006. 1 ; 2n 43 44 Chapter 9. Space-lling curves Figure 9.1: The rst iteration, P1 , of the Hilbert curve. thus P = nlim !1 Pn intuitively has innite length. Referring to Proposition 9.3 and Remark 3.10 as we have done before, we can conrm 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 Thus for m > n we have that sup jPm (t) t 1 1 p2 X 0 i=n p 2 for t 2 [0; 1]: 2n m X1 jPi+1 (t) Pi (t)j Pn (t)j sup 0t1 i=n p 2 =2 n 2i 2 ! 0; as m; n ! 1: Thus (Pn )1 n=1 is uniformly convergent according to the Cauchy criterion for uniform convergence, which states that a sequence, (Pn )1 n=1 , of functions dened on a set E Rn is uniformly convergent if and only if sup jPm (t) t2 E 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 L2 ([0; 1]2 ) = 1 > 0 the claim thus follows. From (5.3), with m = 4 and c = 1=2, we get dimS P = log 4 = 2: 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 dimension 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 denition 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. 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 d2 = p1 d1 = 12 ; 2 46 Chapter 9. Space-lling curves again joined at a right angle. If we let this procedure continue ad inmum we in each iteration Dn , n = 0; 1; 2; : : :, have that every line segment is of length 1 dn = ( p )n 2 and the Heighway dragon, D, is the limit set received, i.e. D = nlim !1 Dn : 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 Heighway 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 p = 2: 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. dimS D = 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 dierent 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 fudgeake (from fudging the von Koch snowake), which is illustrated below. Figure 9.5: The Fudgeake, consisting of three copies of D0 put together. Example 9.6. Changing the angles in the dragon curve from 90 degrees to 120 and using the same starting set, i.e. the unit line, gives us that D10 has side lengths 12 sin 60 = p13 . Each of the 2n line segments in the n:th step has length ( p13 )n and thus intuitively D0 's total length is innite since 2 n p = 1: n!1 3 lim 48 However, Chapter 9. Space-lling curves log 2 2 p = 2 log > 1; log 3 log 3 thus by Remark 3.10 it has innite length. A noticeable fact here is also that dimS D0 is twice as big as dimS C and equal to dimS K . dimS D0 = dimH D0 = 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 claried. 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 dierent dimensions coincide and under which conditions. Then applying the theory to various fractal sets and to investigate how they are connected and dier makes it well worthwhile I think. Leifsson, 2006. 49 50 Chapter 10. Conclusions and nal remarks Bibliography [1] Understanding analysis, Springer-Verlag, New York, 2001. [2] Abbot S., [3] Aigner M., Ziegler G.M., [4] Armstrong M.A., [5] [6] Dimensions and extensions, North-Holland Aarts J.M., Nishiura T., Publishing co., Amsterdam, 1993. Heidelberg, 2004. Proofs from the book, Springer-Verlag, Berlin- Basic topology, Springer-Verlag, New York, 1983. iers L-C., Persson A., Bo Lund, 1988. Crownover R.M., Publishers, 1995. Analys i era variabler, Studentlitteratur, Introduction to fractals and chaos, Jones and Bartlett Edgar G.A., Measure, topology, and fractal geometry, Springer-Verlag, [8] Falconer K., The geometry of fractal sets, Cambridge University Press, [9] Falconer K., Fractal geometry, John Wiley & Sons, Chichester, 1990. Falconer K., Techniques in fractal geometry, John Wiley & Sons, Chich- [7] [10] New York, 1990. Cambridge, 1985. ester, 1997. Fractal geometry, Yale Univer- [11] Frame M., Mandelbrot B., Neger N., [12] Hata M., Kigami J., Yamaguti M., [13] Hurewicz W., Wallman H., [14] rgens H., Maletsky E., Peitgen H-O., Perciante T., Saupe D., Ju sity, 2005. Math. Soc., 1997. Press, 1941. Yunker L., Mathematics of Fractals, Amer. Dimension Theory, Princeton University Fractals for the classroom, Springer-Verlag, New York, 1991. [15] Kreyszig E., Introductory functional analysis with applications, John Wi- [16] Mattila P., Geometry of sets and measures in Euclidean spaces, Cam- ley & Sons, 1978. bridge University Press, Cambridge, 1995. Leifsson, 2006. 51 52 Bibliography [17] Milanov A.V., Zelenyi L.M., [18] Neymark M., [19] [20] [21] Fractal topology and strange kinetics: from percolation theory to problems in cosmic electrodynamics, PhysicsUspekhi 47 (2004), 754. Rogers C.A., bridge, 1970. Rudin W., 1987. Sagan H., Kompendium om konvergens, MAI Linkoping, 2000. Hausdor measures, Cambridge University Press, Cam- Real and complex analysis, 3rd ed., McGraw-Hill, New York, Space-lling curves, Springer-Verlag, New York, 1994. Copyright The publishers will keep this document online on the Internet - or its possible replacement - for a period of 25 years from the date of publication barring exceptional circumstances. The online availability of the document implies a permanent permission for anyone to read, to download, to print out single copies for your own use and to use it unchanged for any non-commercial research and educational purpose. Subsequent transfers of copyright cannot revoke this permission. All other uses of the document are conditional on the consent of the copyright owner. The publisher has taken technical and administrative measures to assure authenticity, security and accessibility. According to intellectual property law the author has the right to be mentioned when his/her work is accessed as described above and to be protected against infringement. For additional information about the Linkoping University Electronic Press and its procedures for publication and for assurance of document integrity, please refer to its WWW home page: http://www.ep.liu.se/ Upphovsratt Detta dokument h alls tillgangligt p a Internet - eller dess framtida ersattare - under 25 ar fr an publiceringsdatum under forutsattning att inga extraordinara omstandigheter uppst ar. Tillg ang till dokumentet innebar tillst and for var och en att lasa, ladda ner, skriva ut enstaka kopior for enskilt bruk och att anvanda det oforandrat for ickekommersiell forskning och for undervisning. Overf oring av upphovsratten vid en senare tidpunkt kan inte upphava detta tillst and. All annan anvandning av dokumentet kraver upphovsmannens medgivande. For att garantera aktheten, sakerheten och tillgangligheten nns det losningar av teknisk och administrativ art. Upphovsmannens ideella ratt innefattar ratt att bli namnd som upphovsman i den omfattning som god sed kraver vid anvandning av dokumentet p a ovan beskrivna satt samt skydd mot att dokumentet andras eller presenteras i s adan form eller i s adant sammanhang som ar krankande for upphovsmannens litterara eller konstnarliga anseende eller egenart. For ytterligare information om Linkoping University Electronic Press se forlagets hemsida http://www.ep.liu.se/ c 2006, Patrik Leifsson Leifsson, 2006. 53

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

### Related manuals

Download PDF

advertisement