DIMENSION OF STRICTLY SELF SIMILAR SETS A Thesis Presented to the Faculty of California State Polytechnic University, Pomona In Partial Fulfllment Of the Requirements for the Degree Master of Science In Mathematics By Santiago Echeverria 2018 SIGNATURE PAGE THESIS: DIMENSION OF STRICTLY SELF SIMILAR SETS AUTHOR: Santiago Echeverria DATE SUBMITTED: Spring 2018 Department of Mathematics and Statistics Dr. Robin Wilson Thesis Committee Chair Mathematics & Statistics Dr. John A. Rock Mathematics & Statistics Dr. Alan Krinik Mathematics & Statistics ii ACKNOWLEDGMENTS Its been a long way since I started as a mere community college student who only thought was nothing more than numbers. I can only thank my family and my friends who were always believing in me and helpingalong the way to this moment. These people have made me into the person I am today and thank them for their support in this long and tiring journey. Especially, my professor Patricia Hale who been very patient with me and helped me with any questions or concerns I have had since I started this paper. iii ABSTRACT In this paper, many concepts and ideas will be presented about fractals and their dimension. There will be a overlook of a few defnitions which classify fractals and what type of dimension they possess. The types of fractals which will be covered are sets which we will classify as strictly and non-strictly self similar sets. Also there will be a comparison of two di˙erent dimensions on a strictly self similar set. iv Contents Signature Page ii Table of Contents vi List of Figures viii 1 Introduction 1 2 Basic Defnitions 4 2.1 Logarithms and Exponential Laws . 4 2.2 Topology . 5 2.3 Functions . 8 2.4 Fractal Defnitions . 12 2.5 Strictly Self Similar Sets . 15 2.5.1 The Cantor Set . 16 2.5.2 The Sierpinski Triangle . 16 2.5.3 The Von Koch Curve . 19 3 Non-Similar Fractals 23 3.1 The Julia Sets . 23 v 3.2 The Mandelbrot Set . 25 4 Dimension 30 4.1 Topological Dimension . 31 4.2 Similarity Dimension . 32 4.3 Hausdor˙ Dimension . 35 4.3.1 Hausdor˙ Measure . 35 5 Comparing Dimension 42 References 49 vi List of Figures 1.1 The Cantor Set [2] . 2 1.2 The Cantor Set . 3 2.1 Similar Triangles [5] . 12 2.2 Diagram of a Fern [4] . 14 2.3 A Shoreline [20] . 14 2.4 Waclaw Sierpinski [16] . 17 2.5 The Sierpinski Triangle [15] . 18 2.6 Open set under IFS . 19 2.7 The Von Koch Curve [7] . 21 2.8 Helge Von Koch [11] . 21 2.9 Open Trapezoid under IFS . 22 3.1 The Mandelbrot Set [17] . 25 3.2 Main Cardioid, Primary bulbs, Spokes, and Main antennas [6] . 27 3.3 4th primary bulb on the Mandelbrot Cactus [6] . 28 3.4 Smallest antenna on the 4th primary bulb [6] . 28 3.5 Fibonacci Numbers Found in the Mandelbrot set [6] . 29 4.1 Unit Cube [1] . 33 vii 4.2 The Sierpinski Triangle . 36 4.3 Closed balls covering boundary of unit square. 38 4.4 The Hausdor˙ dimension is the value where jump discontinuity oc- curs [3] . 39 n 5.1 The sets En = Vm where n = 0; 1; 2 . 45 i 5.2 C \ Vj where i = 0; 1; 2 . 46 viii Chapter 1 Introduction The understanding of a set containing some similar pat- tern or relation throughout its structure is the study of fractals. Fractals appear in many shapes and forms in our own world such as in plants, animals, and in math- ematical objects. These fractals are becoming more and more popular by mathematicians who wish to fnd their secrets which are hidden in plain sight. These secrets could be just how two fractals relate or how to product Georg Cantor [18] such an object. Before we start to give answers to these topics we need to know what fractals are and how they are constructed. When Georg Cantor (1845-1918) frst started to lay the foundations for modern day set theory, he introduced one of the most important fractals in his paper pub- lished in 1883 called the Cantor Set. The Cantor Set is a fractal and is an infnite set of points which are usually considered to lie in the closed interval [0,1] or unit interval. The mannerin which we constructthe Cantor Set is by an infniteprocess 1 that follows the following steps: • Step 0: Consider the closed interval F0 = [0; 1]. rd • Step 1: Remove the middle 3 of the interval [0; 1] and so we are left with 1 2 [0; 3 ][[ 3 ; 1]. Note that at each of these two intervals are similarto the original 1 with similarity constant 3 and call this new set F1. rd • Step 2: Remove the middle 3 from each interval in F1 so that we are left 1 2 12 7 8 with [0; 9 ] [ [ 9; 3 ] [ [ 3; 9 ] [ [ 9 ; 1]. Again, each of these intervals is similar to 1 the original interval [0; 1] with the similarity constant 32 and label this new set F2. rd k • Step k: Remove the middle 3 from each interval in Fk−1 which leaves 2 1 intervals of length 3k ; this stage is called the kth stage or Fk. Figure 1.1: The Cantor Set [2] At each stage, we can count how many intervals there are and fnd their lengths k 1 which is 2 intervals with length of 3k at the kth stage. It is standard to name each stage of a fractal by F0;F1;F2; :::; Fk where k 2 N and we will do so regularly in this paper. Note these stages are just a fnite number of representations in the 2 process of construction the Cantor Set and the real Cantor Set is obtained when k tends to infnity and we denote it as F . We can also take the intersection of all 1 such Fk to obtain the Cantor Set such as F = \k=0Fk. 1 Example 1. Consider the Cantor Set F . Scale F by 3 , then the resulting set is a smaller and yet a similar set to the Cantor Set. This set is an exact copy of the left hand subset of the Cantor Set. See the fgure below. Figure 1.2: The Cantor Set Now that we have seen what a fractal could be and how it could be generated, we will now go into the theory of fractals. This theory requires for the reader to recall some basic defnitions and theorems which are stripped bare so only the important material is given. 3 Chapter 2 Basic Defnitions 2.1 Logarithms and Exponential Laws In this paper we will face many concepts which involve fractals. These fractals are complex in nature but our focus will be mainly on the dimension of such objects. Once we give several di˙erent defnitions of dimension we will face how to calculate these dimensions which involves using techniques of logarithms and exponential laws. Before we undertake these calculations we need some review. Defnition 1. If b and x are positive real numbers and y is a real number then, y y = logb x is equivalent to b = x: We also read this as log base b of x. In the defnition above, we call the equality y = logb x the logarithmic form and the following equality, by = x, the exponential form. Next we will recall some useful properties of logarithms. Proposition 1 (Wolfram). Suppose a, b, and c are positive real numbers and p any real number then 4 1: log c ab = log c a + log c b a 2: log c b = log c a − log c b p 3: logc a = p logc a log a 4: log c a = log c 1 5: logc a = loga c The above properties can all be verifed by the defnition and properties of exponents. The following collection of properties which we will call proposition 2 can be proven by the properties in proposition 1. Since the material is review, which the reader has seen, we do not prove these properties but insteadleave them as exercises for the reader. Proposition 2. (Wolfram) Suppose a, b, and c are positive real numbers and p is any real number then 1: logc 1 = 0 2: logc c = 1 p 3: logc c = p 4: log c a log a b = log c b We remind the reader these properties will be helpful when we solving for the actual dimension of a fractals in the later chapters. 2.2 Topology Before we establish some conceptsof functions we need some other useful defnitions from Topology. We will not go into too much theory on the subject but the main theorem we will like to touch on is the Heine-Borel Theorem. 5 Defnition 2. A topology on a set X is a collection τ of subsets of X having the following properties: 1. ; and X are in τ 2. The union of the elements of any sub-collection of τ is in τ 3. The intersection of the elements of any fnite sub-collection of τ is in τ The set X for which a topology τ has been specifed is called a topological space. Defnition 3. If X is a topological space with topology τ, then the subset V of X is said to be an open set of X if V belongs to the collection τ. Defnition 4. A subset A of a topological space X is said to be closed if the set X − A is open.