Mathematics and Statistics 6(3): 25-33, 2018 http://www.hrpub.org DOI: 10.13189/ms.2018.060301 Hausdorff Measures and Hausdorff Dimensions of the Invariant Sets for Iterated Function Systems of Geometric Fractals 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 Copyright©2018 by authors, all rights reserved. Authors agree that this article remains permanently open access under the terms of the Creative Commons Attribution License 4.0 International License Abstract In this paper, we discuss Hausdorff measure xn+1 a b xn e and Hausdorff dimension. We also discuss iterated = + yn+1 c d yn f function systems (IFS) of the generalized Cantor sets and higher dimensional fractals such as the square fractal, the where a,, b c 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 w,, w w as a set of affine invariant sets for IFS of these fractals. 1 2 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 F( A )= w ( A ), 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 0<β < 1. Let p,,, p p be points in the Any fractal has some infinitely repeating pattern. When 1 2 N crating such fractal, repetition of a certain series of steps is plane. Let =β − + where x Ai ()(), p p pi p i p = necessary which create that pattern. Iterated Function y System is another way of generating fractals. It is based on and i = 1,2,3, ,N . The collection of functions taking a point or a figure and substituting it with several other identical ones. Iterated function system represents an {,,,}AAA1 2 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 |Ui |≤ δ ∞ ∞ 1. 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). ∈ n n Definition 2.2. [7] Let E R be a Borel set with S :R → R Let 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 x, y ∈Rn , 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 t s λδ -cover} ≤ for any positive δ After t −s H δ (E) H δ (E) . ∞ ∞ ∞ δ s s s s t Here | S(Ui ) | ≤ | λUi | = λ |Ui | (E) = ∞, ∑ ∑ ∑ taking limits, we see that 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 = S (F) then each |U | ≤ δ since |U |≤ δ , so i i i = 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.