Hausdorff Measures and Hausdorff Dimensions of the Invariant Sets for Iterated Function Systems of Geometric Fractals

Mathematics and Statistics 6(3): 25-33, 2018 http://www.hrpub.org DOI: 10.13189/ms.2018.060301

Md. Jahurul Islam1,*, Md. Shahidul Islam1, Md. Shafiqul Islam2

1Department of Mathematics, University of Dhaka, Bangladesh 2School of Mathematics and Computational Science, University of Prince Edward Island, Canada

Abstract In this paper, we discuss Hausdorff measure  xn+1   ba  xn   e  and Hausdorff dimension. We also discuss iterated   =    +    yn+1   dc  yn   f  function systems (IFS) of the generalized Cantor sets and higher dimensional fractals such as the square fractal, the where ,, cba and d control rotation and scaling, while Menger sponge and the Sierpinski tetrahedron and show e and f control linear translation. the Hausdorff measures and Hausdorff dimensions of the Now we consider ,, www as a set of affine invariant sets for IFS of these fractals. 21 N linear transformations, and let A be the initial geometry. Keywords Hausdorff Measure, Hausdorff Dimension, Then a new geometry can be represented by Invariant Set, Iterated Function System N = AwAF ,)()(  i i=1 where F is known as the Barnsley-Hutchinson operator [3, 4]. 1. Introduction We can also define iterated function system as follows: Let β << .10 Let ,,, ppp be points in the Any fractal has some infinitely repeating pattern. When 21 N crating such fractal, repetition of a certain series of steps is plane. Let β +−= where  x  i ppppA ii ,)()( p =   necessary which create that pattern. Iterated Function  y System is another way of generating fractals. It is based on and i =  N.,,3,2,1 The collection of functions taking a point or a figure and substituting it with several other identical ones. Iterated function system represents an 21  AAA N },,,{ is called an iterated function system extremely versatile method for conveniently generating a [5]. wide variety of useful fractal structures [1]. In Section 2, we present a brief review on Hausdorf In this paper, we study the Cantor set and formulate measure, their properties and Hausdorff dimension. iterated function system with probabilities of the generalized Moreover, we present Hausdorff dimension of the invariant Cantor sets and also show their invariant measures using set for contracting maps. In Section 3, we present Markov operator and Barnsley-Hutchison multifunction [2]. Hausdorff measures and Hausdorff dimensions of the We formulate iterated generalized cantor sets. Hausdorff measures and Hausdorff function system of two dimensional the square fractal dimensions of the invariant sets for IFS of the generalized and the three dimensional fractals such as the Menger cantor sets are presented in Section 4. In Section 5, we sponge, the Sierpinski tetrahedron. We also discuss present some examples of two dimensional fractals and Hausdorff measures and Hausdorff dimensions of the determine Hausdorff measures and Hausdorff dimensions invariant sets for iterated function systems of these fractals. of the invariant sets for IFS of these fractals. In Section 6, The Iterated Function System is based on the application we present examples of Hausdorff measures and Hausdorff of a series of Affine Transformations. An Affine dimensions of the invariant sets for IFS of three Transformation is a recursive transformation of the type [3] dimensional fractals. 26 Hausdorff Measures and Hausdorff Dimensions of the Invariant Sets for Iterated Function Systems of Geometric Fractals

2. Preliminaries (i) (Null Empty set) s φ = H ( ) 0. (ii) (Monotonicity) If E ⊆ F, then 2.1. [6] Hausdorff Measure s s Let U be any non-empty subset of the n-dimensional H (E) ≤ H (F). n Euclidean space R . The diameter of U is defined as (iii) (Countable Subadditivity) = − ∈ |U |: sup{| x y |: x, y U}. ∞ ∞ s ( E ) ≤  i ∑ Here we will use the Euclidean metric: i=1 i=1 s (E ). 2 2 2 1/ 2 H H i | x − y |:= ((x1 − y1) + (x2 − y2 ) ++ (xn − yn ) ) . Proof: The proof can be found in [8]. However, as will be shown shortly, we may use any L p Theorem 2.5.2. (Countable Additivity) n ∞ s metric. If E ⊆ R , and a collection {U i}i∈I satisfies Let be a countable collection of disjoint – {Ei}i=1 H the following conditions: measurable subsets of X. Then we have |U |≤ δ 1. i for each ∈ ∞ ∞ i I; s ( Ei ) = ∑ 2. E ⊆ U ,  s i∈I i i=1 i=1  H H (Ei ). then we say the collection is a δ -cover of E. We may Proof: The proof can be found in [9]. assume the collection is always countable. n Theorem 2.5.3. [7] (Scaling Property). Definition 2.2. [7] Let E ∈R be a Borel set with n → n Let S :R R be a similarity mapping, and any {U } ∈ a δ -cover for it. Given any s > 0, we define i i I | S(x) − S(y) |≤ λ | x − y | the δ -approximating s-dimensional Hausdorff measure λ > 0 such that for any n n s n → as follows x, y ∈R , E ∈R , H δ :R R where λ is the scaling factor. If ∞ s (S(E) = λs s (E). s s then δ (E) = inf{∑|Ui | :{Ui}i∈I H H i=1 forms a ∞ δ − E. H Proof: Let {Ui}i=1 be a cover of Then δ ∞ λδ -cover for E}. {S(Ui )}i=1 is a -cover of S(E). Thus we see that, δ → If 0, then we have a more succinct formula given s > 0 : s = ( s ∞ ∞ (E) lim δ (E)) s s {U } H δ →0 H = i i=1 forms a H λδ (S(E)) inf{∑ | S(Ui ) | : i=1 gives the s-dimensional Hausdorff measure of E. λδ -cover} n n Definition 2.3. A mapping S : R → R is a ∞ ∞ ∞ s s s s similarity with ratio r > 0 if Here ∑ | S(Ui ) | ≤ ∑ | λUi | = λ ∑ |Ui | i=1 i=1 i=1 − = − | S(x) S(y) | r | x y | . We obtain Definition 2.4. Let (M ,d) be a metric space. A s s s λδ (S(E)) ≤ λ δ (E). mapping f : M → M is a contraction or contractor H H with the property that there is some nonnegative real So, letting δ → 0 gives that number such that for all x and y in M , s ≤ λs s ≤ H (S(E)) H (E). d( f (x), f (y)) kd(x, y). The smallest such value of k is called the Lipschitz constant of f . Contractive maps are sometimes called 1 Replacing λ by and E by S(E) gives the Lipschitzian maps. λ opposite inequality, as required. 2.5. Properties of Hausdorff Measure Theorem 2.5.3. [7] (Scaling Property). Theorem 2.5.1. (Outer Measure) The following are true S :Rn → Rn for any metric space Let be a similarity mapping, and any

Mathematics and Statistics 6(3): 25-33, 2018 27

| S(x) − S(y) |≤ λ | x − y | s < ∞ > t = λ > 0 such that for any 1. If (E) and t s, then (E) 0. H H ∈ n x, y R , where λ is the scaling factor. If Proof: Equation (1) shows that − ∈ n s = λs s t (E) ≤ δ t s s (E) for any positive δ. E R , then (S(E) (E). H δ H δ H H The result follows after taking limits, since ∞ Proof: Let {U } be a δ − cover of E. Then i i=1 s (E) < ∞. ∞ λδ H {S(Ui )}i=1 is a -cover of S(E). Thus we see that, given s > 0 : s > < t = ∞ 2. If (E) 0 and t s, then (E) . ∞ ∞ H H s s {U } = i i=1 forms a H λδ (S(E)) inf{∑ | S(Ui ) | : Proof: Equation (1) shows that i=1 1 λδ -cover} t ≤ s for any positive δ After t −s H δ (E) H δ (E) . ∞ ∞ ∞ δ s s s s Here | S(U ) | ≤ | λU | = λ |U | t = ∞ ∑ i ∑ i ∑ i taking limits, we see that (E) , since i=1 i=1 i=1 H We obtain 1 lim = ∞ and s s s δ →0 t −s λδ (S(E)) ≤ λ δ (E). δ H H s s > = (E) 0. So, letting δ → 0 gives that limδ →0 H δ (E) H One immediate consequence of these observation is that s s s H (S(E)) ≤ λ H (E). ∞ if s < dim H (E) 1  λ s Replacing by and E by S(E) gives the (E) = 0 if s > dim H (E) λ  opposite inequality, as required. finite, nonzero if s = dim (E)  H 2.5.4. [7] Hausdorff Dimension Hausdorff Dimension of the Invariant set for Let be a given set. Note that s decreases as E H δ (E)) Contraction Maps increases. This means that s also decreases with Let S , S ,, S be contractions. A subset F of s H (E) 1 2 N X is called invariant for the transformation Si if s increasing. If t > s and {U i} is a δ -cover of E, N t −s t −s F = Si (F) then each |Ui | ≤ δ since |Ui |≤ δ , so  i=1 ∞∞ t= ts− s ∑∑||Ui (||||) UU ii In the case where Si :X → X are similarities with ii=11=

∞∞Lipschitz constants Li for i =1,2,, N respectively, ts−− s ts s a theorem proved by M. Hata (Theorem 10.3 of [9] and ≤≤∑∑(δδ ||)UUii ||. ii=11= Proposition 9.7 of [6]) allows us to calculate the Hausdorff

After taking the infima over all δ -covers, we can easily dimension of the invariant set for S1,S2 ,,S N . see that Namely, if we assume that F is an invariant set for the t t −s s similarities S , S ,, S and S (F) ∩ S (F) = φ ≤ δ (1) 1 2 N i j H δ (E) H δ (E) for i ≠ j, then Let δ → Then t → if s is finite. = 0. H δ (E) 0 H (E) dim H F s, where s is given by t Also if δ is bounded and finite then N H s Li = 1. (2) s ∑ → ∞ i=1 H δ (E) . Two applications of equation (1) should be noted: 3. Hausdorff Measures and Hausdorff

28 Hausdorff Measures and Hausdorff Dimensions of the Invariant Sets for Iterated Function Systems of Geometric Fractals

Dimensions of the Generalized 1 Hausdorff dimension of the Cantor middle Cantor Sets 2m −1 (2 ≤ m < ∞), set is 1 Let C1/3 be the Cantor middle set. The set C1/3 log m 3 dim H C1/(2m−1) = . log (2m −1) 1 splits into a left part (C1/ 3 )L = C1/ 3 [0, ] and a right log m Since dim C − = <1, 3 H 1/(2m 1) log (2m −1) 2 part (C ) = C [ ,1]. s (C ) = 0 for 2 ≤ m < ∞. 1/ 3 R 1/ 3 3 H 1/(2m−1) Hence the Hausdorff measure of the generalized Cantor sets Clearly, both parts are geometrically similar to C 1/3 is zero. 1 but scaled by a ratio and C = (C )  (C ) 3 1/ 3 1/ 3 L 1/ 3 R with this union disjoint. Thus by the scaling property we have s = s + s H (C1/ 3 ) H ((C1/ 3 )L ) H ((C1/ 3 )R ) s s  1  s  1  s =   (C1/ 3 ) +   (C1/ 3 )  3 H  3  H s 1 (assume 0 < (C1/ 3 ) < ∞ when s = dimH F ) Figure 1. Construction of the Cantor middle set as IFS H 2m −1 Now cancelling s form both sides, we have H (C1/ 3 ) s s  1   1  −s 1 =   +   ⇒1 = 2×3 . 4. Hausdorff Measures and Hausdorff  3   3  Dimensions of the Invariant Sets for log 2 ∴s = = 0.631 IFS of the Generalized Cantor Sets log3 Iterated Function System of the Generalized Cantor Sets: 1 Let = Let ρ be a complete separable Thus the Hausdorff dimension of the Cantor middle X [0,1]. (X , ) 3 metric space. If wk :X → X is a function which is set is = dimH C1/ 3 0.631. defined by 1 1 x 2(k −1) Since the Cantor middle set is in R , w (x) = + , m ≥ 2 3 k 2m −1 2m −1 dim C = 0.631 < 1, s (C ) = 0. 1 H 1/3 H 1/ 3 with contracting factor L = ,(2 ≤ m < ∞) for 1 2m −1 Hence the Hausdorff measure of the Cantor middle k =1,2, ,m respectively, then the family 3  set is zero. {wk :k = 1, 2,, m} is called an iterated function Similarly, we can show that the Hausdorff dimension of system of the generalized Cantor sets (IFSGCS) which is 1 1 the Cantor middle set is denoted by the Cantor middle ,(2 ≤ m < ∞) sets. 5 2m −1 dimH C1/ 5 = 0.682. Let F be an invariant set for IFS of the Cantor middle s 1 Since dimH C1/ 5 = 0.682 <1, (C1/ 5 ) = 0. set. If F ⊆ R, then w (F) ∩ w (F) = φ for H − i j We obtain the Hausdorff measure of the Cantor middle 2m 1 1 1 i ≠ j, and L = for i = 1, 2,, m. set is zero. In general, we can show that the i − 5 2m 1 Now from (2), we have

Mathematics and Statistics 6(3): 25-33, 2018 29

s s  1  ⇒ 3 = 5 ⇒ slog3 = log5 m ⋅   = 1 ⇒ s log( 2m-1) = log m  2m −1 log5 ∴s = =1.46. log m log3 ∴ s = ,(2 ≤ m < ∞). log (2m −1) Thus the Hausdorff dimension of the invariant set for Thus the Hausdorff dimension of the invariant set for IFS of IFS of the box fractal is dim B =1.46. the generalized Cantor sets is H 2 log m Since the box fractal is in R , and dim F = ,(2 ≤ m < ∞). H = < s = log (2m −1) dimH B 1.46 2, H (B) 0. Hence the Hausdorff measures of the generalized Cantor sets Hence the Hausdorff measure of the invariant set for IFS is zero. of the box fractal is zero. 5. Hausdorff Measures and Hausdorff Dimensions of the Invariant Sets for IFS of Two Dimensional Fractals

5.1. Hausdorff Measure and Dimension of the Invariant Set for IFS of the Box Fractal

Let B be an invariant set for IFS of the box fractal which is defined by the following 1 1 1 2 1 w1 (x, y) = ( x, y), w2 (x, y) = ( x + , y), B 3 3 3 3 3 0 1 1 2 w (x, y) = ( x, y + ), 3 3 3 3 1 2 1 2 w (x, y) = ( x + , y + ), 4 3 3 3 3 1 1 1 1 w (x, y) = ( x + , y + ) 5 3 3 3 3 Now we have

ρ(w1(x, y),w1(x′, y′))

1 2 1 2 B (1st Iteration) = (x − x′) + (y − y′) 1 9 9 1 = (x − x′)2 + (y − y′)2 3 1 = ρ((x, y),(x′, y′)) 3

It follows that w1, w2 , w3 , w4 and w5 are

2 1 contraction on R with L = . 3 5 2 1 If ⊆ then w (B) = φ and L = for B nd B R ,  i i 2 (2 Iteration) i=1 3 each i = 1,2,3,4,5. Now from (2), we have Figure 2. onstruction of the box fractal as IFS s s s s s  1   1   1   1   1    +   +   +   +   =1 5.2. Hausdorff Measure and Dimension of the Invariant  3   3   3   3   3  Set for IFS of the Square Fractal (Using the Cantor

30 Hausdorff Measures and Hausdorff Dimensions of the Invariant Sets for Iterated Function Systems of Geometric Fractals

1 Middle Set) 3 Let S be an invariant set for IFS of the square fractal 1 (using the Cantor middle set) which is represented by the 3 following:

1 1 1 2 1 S2 nd w (x, y) = ( x, y), w (x, y) = ( x + , y) (2 Iteration) 1 3 3 2 3 3 3 1 1 1 2 Figure 3. Construction of the square fractal (using the Cantor middle w3 (x, y) = ( x, y + ), 3 3 3 3 set) as IFS 1 2 1 2 w (x, y) = ( x + , y + ) 2 4 3 3 3 3 Since the square fractal is in R , and We have dimH S =1.26 < 2, ρ ′ ′ s (w1(x, y),w1(x , y )) = H (S) 0. 1 2 1 2 Hence the Hausdorff measure of the invariant set for IFS of = (x − x′) + (y − y′) 9 9 the square fractal is zero.

1 2 2 = (x − x′) + (y − y′) 5.3. Hausdorff Measure and Dimension of the Invariant 3 Set for IFS of the Square Fractal (Using the Cantor 1 = ρ ′ ′ 1 ((x, y),(x , y )) Middle Set) 3 5 It follows that w1, w2 , w3 and w4 are contraction Let F be an invariant set for IFS of the square fractal 2 1 on R with L = . which is defined by the following: 3 1 1 1 2 1 4 w (x, y) = ( x, y), w (x, y) = ( x + , y), 1 1 2 If S ⊆ R2 , then w (S) = φ and L = for 5 5 5 5 5  i i i=1 3 1 1 2 w (x, y) = ( x, y + ), each = Now from (2), we have 3 i 1,2,3, 4. 5 5 5 s s s s 1 4 1  1   1   1   1  = +   +   +   +   =1 w4 (x, y) ( x , y),  3   3   3   3  5 5 5 ⇒ s = ⇒ = 1 1 4 3 4 slog3 log4 w5 (x, y) = ( x, y + ), log 4 5 5 5 ∴s = =1.26. 1 2 1 2 log3 w6 (x, y) = ( x + , y + ), Thus the Hausdorff dimension of the invariant set for IFS of 5 5 5 5 1 2 1 4 the square fractal is dimH S =1.26. w (x, y) = ( x + , y + ), 7 5 5 5 5 1 4 1 2 w (x, y) = ( x + , y + ), 8 5 5 5 5 1 4 1 4 w (x, y) = ( x + , y + ) 9 5 5 5 5 Now we obtain

S 0 S1 (1st Iteration)

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ρ ′ ′ (w1(x, y),w1(x , y )) S2 (2nd Iteration) 1 1 2 1 2 Figure 4. Construction of the square fractal (using the Cantor middle = (x − x′) + (y − y′) 25 25 set) as IFS 5 1 = (x − x′)2 + (y − y′)2 5 6. Hausdorff Measures and Hausdorff 1 Dimensions of the Invariant Sets for = ρ((x, y),(x′, y′)) 5 IFS of Three Dimensional Fractals

It follows that w1, w2 ,w3,,w8 and w9 are 6.1. Hausdorff Measure and Dimension of the Invariant 2 1 contraction on R with L = . Set for IFS of the Menger Sponge 5 9 Let M be an invariant set for IFS of the Menger sponge 1 If S ⊆ R 2 , then w (S) = φ and L = for which is represented by the following:  i i i=1 5  1 1 1  w (x, y, z) =  x, y, z, each i = 1,2,3,,9. 1  3 3 3  Now from (2), we have  1 1 1  s w (x, y, z) =  (x +1), y, z,  1  s 2 9⋅  =1⇒ 5 = 9 ⇒ slog5 = log9  3 3 3  5    1 1 1  log9 w3 (x, y, z) =  x, (y +1), z, ∴s = =1.36.  3 3 3  log5  1 1 1  Thus the Hausdorff dimension of the invariant set for w (x, y, z) =  (x +1), y, (z +1), 4  3 3 3  IFS of the square fractal is dimH F = 1.36. 2  1 1 1  Since the square fractal is in R , and w (x, y, z) =  (x + 2), y, z, 5  3 3 3  dim H S = 1.36 < 2, s  1 1 1  = w6 (x, y, z) =  x, (y + 2), z, H (S) 0.  3 3 3  Hence the Hausdorff measure of the invariant set for IFS  1 1 1  of the square fractal is zero. w (x, y, z) =  x, y, (z + 2), 7  3 3 3   1 1 1  w8 (x, y, z) =  (x +1), (y + 2), z,  3 3 3   1 1 1  w (x, y, z) =  (x +1), y, (z + 2), 9  3 3 3   1 1 1  w10 (x, y, z) =  x, (y +1), (z + 2),  3 3 3   1 1 1  S S st 0 1 (1 Iteration) w11 (x, y, z) =  (x + 2), (y +1), z,  3 3 3   1 1 1  w12 (x, y, z) =  (x + 2), y, (z +1),  3 3 3   1 1 1  w (x, y, z) =  x, (y + 2), (z +1), 13  3 3 3   1 1 1  w14 (x, y, z) =  (x + 2), (y + 2), z,  3 3 3 

32 Hausdorff Measures and Hausdorff Dimensions of the Invariant Sets for Iterated Function Systems of Geometric Fractals

 1 1 1  w (x, y, z) =  (x + 2), y, (z + 2), 15  3 3 3   1 1 1  w16 (x, y, z) =  x, (y + 2), (z + 2),  3 3 3   1 1 1  w (x, y, z) =  (x + 2), (y + 2), (z +1), 17  3 3 3 

 1 1 1  w (x, y, z) =  (x + 2), (y +1), (z + 2), M 18  3 3 3  0  1 1 1  w (x, y, z) =  (x +1), (y + 2), (z + 2), 19  3 3 3   1 1 1  w (x, y, z) =  (x + 2), (y + 2), (z + 2) Now 20  3 3 3  we have

ρ(w1 (x, y, z),w1 (x′, y′, z′))

1 1 1 = (x − x′)2 + (y − y′)2 + (z − z′)2 9 9 9 M1 (1st Iteration) 1 = (x − x′)2 + (y − y′)2 + (z − z′)2 3 1 = p((x, y, x),(x′, y′, z′)) 3

It follows that w1, w2 ,, w19 and w20 are

3 1 contraction on R with L = . 3 20 3 1 If M ⊆ R , then w (M ) = φ and L = for  i i i=1 3 M nd each i = 1,2,,20. 2 (2 Iteration) Now from (2), we have Figure 5. Construction of the Menger sponge as IFS s  1  20⋅  =1⇒ 3s = 20 ⇒ slog3 = log20  3  6.2. Hausdorff Measure and Dimension of the Invariant Set for IFS of the Sierpinski Tetrahedron log 20 ∴s = = 2.726. log3 Let T be an invariant set for IFS of the Sierpinski tetrahedron which is defined by the following: Thus the Hausdorff dimension of the invariant set for x y (z − 3) IFS of the Menger sponge is dimH M = 2.726. Since w (x, y, z) = ( , , ), 1 2 2 2 the Menger sponge is in R3, and x y 2 z 1 dim H M = 2.726 < 3, w2 (x, y, z)= ( , + , − ), 2 2 3 2 2 3 s (M ) = 0. H x 1 y 1 z 1 w (x, y, z)= ( − , − , − ), Hence the Hausdorff measure of the invariant set for IFS 3 2 2 2 of the Menger sponge is zero. 2 6 2 3 x 1 y 1 z 1 w4 (x, y, z)= ( + , − , − ) 2 2 2 6 2 2 3 We have

Mathematics and Statistics 6(3): 25-33, 2018 33

ρ(w1 (x, y, z),w1 (x′, y′, z′)) 1 1 1 = (x − x′)2 + (y − y′)2 + (z − z′)2 4 4 4 1 = (x − x′)2 + (y − y′)2 + (z − z′)2 2 1 = p((x, y, x),(x′, y′, z′)) 2

It follows that w1, w2 , w3 and w4 are contraction on T2 (2nd Iteration) 3 1 R with L = . 2 Figure 6. Construction of the Sierpinski tetrahedron as IFS 4 3 1 If T ⊆ R , then w (T) = φ and L = for  i i 2 i=1 7. Conclusions each i = 1,2,3, 4. From equation (2), we get s We observe that the Hausdorff dimensions of the  1  invariant sets for IFS of the geometric fractals are different. 4⋅  =1⇒ 2s = 4 ⇒ slog2 = log4  2  But the Hausdorff measures of the invariant sets for IFS of the geometric fractals are zero as the Hausdorff dimensions log 4 ∴s = = 2. are less than the Euclidean dimensions of the log 2 corresponding fractals. Thus the Hausdorff dimension of the invariant set for IFS of the Sierpinski tetrahedron is dimHT = 2. Since the Sierpinski tetrahedron is in R3, and REFERENCES [1] H. O. Peitger, H. Jungeus and D. Sourpe, Chaos and dimHT = 2 < 3, Fracatls: New frontier of science, Springer, New York, s = 1992. H (T) 0. Hence the Hausdorff measure of the invariant set for IFS [2] M. J. Islam, M. S. Islam, Invariant Measures for Iterated of the Sierpinski tetrahedron is zero. Function Systems of Generalized Cantor Sets, German J. Ad. Math. Sci., Vol. 1(2) (2016), 31-47. [3] M. F. Barnsley, Fractals Everywhere, Academic Press, Massachusetts, 1993. [4] J. E. Hutchison, Fractal and self-similarity, Indiana Math. J. 30(1981), 713-743. [5] R. L. Devaney, a First Course in Chaotic Dynamical Systems, Boston University, Addison-Wesley, West Views Press, 1992. [6] K. J. Falconer, Fractal Geometry: Mathematical Foundation and Applications, Wiley, New York, 1990. [7] N. Loughlin, A. Wallis, Fractal Geometry MT4513/5813, 2006. Online available fromhttps://www.students.ncl.ac.u T k/n.j.loughlin/fractal2.pdf 0 T1 (1st Iteration) [8] P. Lertchoosakul, Introduction of Hausdorff Measure and Dimension, Dynamics Learning Seminar, Liverpool, 2012. Online available from pojlertchoosakul.webs.com/Introduc tionToHausdorffDimension.pdf [9] M. Hata, On the Structure of Self-similar Sets, Japan J. Appl. Math., 2 (1985), 381-414.