THE FRACTAL DIMENSION OF THE WEIERSTRASS TYPE FUNCTIONS by ..-' LEE TI^N WAH • Thesis Submitted to the Faculty of the Graduate School of The Chinese University of Hong Kong (Division of Mathematics) In partial fulfillment of the requirements for the Degree of Master of Philosophy July, 1998 f^ ^S&-.•<:"^^^^¾!!^^^^^ _ £^^ H \ 9 JULJ^HI k "uN!VER^"""""“^/^I ^^SUBRARY SYST^^ 'X^^^ 撮要 本論文的目的是計算一些與Weierstrass函數有關的 圖形的分形維數0 Weierstrass函數是一個眾所周知 連續但不可微分的函數。關於這個函數的圖形， 我們知道它的盒維數但是它的Hausdorff維數則仍 是一個公開而具有挑戰性的問題。在論文中，我 們運用兩種方法來計算Weierstrass函數圖形的 Hausdorff維數，他們是質量分佈準則和位勢理論方 法。首先我們構造一個量度並運用質量分佈準則 來計算它的Hausdorff維數的下限。然後我們加入 一個參數9在?616『31『355函數裏並透過位勢理論方 法，則對幾乎所有參數e我們可以得出它的 Hausdorff維數。自仿射曲線是另一種圖形由迭代 函數系統構造而成。這種圖形可分為兩種類別。 第一種是可微分的自仿射曲線，他們是易於計算 的。另一種是分形的自仿射曲線，它們是我們主 要關注的課題。正如Weierstrass函數，它的 Hausdorff維數仍不容易計算。我們運用位勢理論 方法來計算它的Hausdorff維數。最後，我們引進 Dekking[ll]的 recurrent M � Recurrent 集的表示法是將 代數和分析聯系起來，並且是一種有效的方法來 描述和製造space-filling 圖形和一些相似於 Weierstrass函數的圖形。 The Fractal dimension of the Weierstrass type functions 1 Abstract Our aim in the thesis is to estimate the fractal dimension of certain graphs related to the Weierstrass functions. The Weierstrass function is the best-known example of a continuous but nowhere differentiable function. The box dimension of the graph is well-known, however, its Hausdorff dimension is a still challenging open problem. In this thesis, two methods are used to investigate the Hausdorff dimen- sion of the Weierstrass function[5]. They are the mass distribution principle[l and the potential theoretic method[l]. By using the mass distribution principle, a measure is constructed to estimate the lower bound ofthe Hausdorff dimension[5 . The potential theoretic method introduces a new parameter and gives simple for- mula to express the Hausdorff dimension for，，almost all" cases in the parameter space[7]. Self-affine curves are constructed by Iterated function system (IFS) and are another type of curves analogous to the Weierstrass function. We discuss the regularity of such curves. The differentiable case is easy. For the fractal case, the Hausdorff dimension is difficult to obtain. Falconer[3] use the potential theoretic method to study such curve and its Hausdorff dimension. Finally, the recurrent set[ll] is introduced which is primarily based on interplaying between algebra and analysis. Its description is a powerful method of describing and generating space-filling curves and graphs of nowhere differentiable continuous functions like the Weierstrass function. The Fractal dimension of the Weierstrass type functions 2 ACKNOWLEDGMENTS I wish to express my sincere and deepest gratitude to my supervisor, Profes- sor K. S. Lau, for introducing this topic to me. With his inspired guidance and discussions and achieve the thesis, I have increased my knowledge in this frontier of mathematics. LEE TIN WAH The Chinese University of Hong Kong August 1998 Contents 1 Introduction 5 2 Preliminaries 8 2.1 Box dimension and Hausdorff dimension 8 2.2 Basic properties of dimensions 9 2.3 Calculating dimensions 11 3 Dimension of graph of the Weierstrass function 14 3.1 Calculating dimensions of a graph 14 3.2 Weierstrass function 16 3.3 An almost everywhere argument 23 3.4 Tagaki function 26 4 Self-afRne mappings 30 4.1 Box dimension of self-affine curves 30 4.2 Differentability of self-affine curves 35 4.3 Tagaki function 42 4.4 Hausdorff dimension of self-affine sets 43 3 The Fractal dimension of the Weierstrass type functions 4 5 Recurrent set and Weierstrass-like functions 56 5.1 Recurrent curves 56 5.2 Recurrent sets 62 5.3 Weierstrass-like functions from recurrent sets 64 Bibliography Chapter 1 Introduction The focus of the thesis is to discuss a variety of interesting fractals occuring as graphs of functions. The usual way of measuring a fractal is by some form of 'dimension'. The ,dimension, refers to the box dimension and Hausdorff di- mension. The former one is conceptually simple and easy to use. Its popularity is largely due to its relative ease of mathematical calculation and empirical es- timation. However, the latter one is more desirable mathematically dues to the property of countable stability. But it is harder to get the lower estimates. There are two techniques to finding the Hausdorff dimension of a fractal set. They are the mass distribution principle[l] and the potential theoretic method[l]. These techniques are used repeatedly in this thesis. The Weierstrass function 00 f{x) = J2 X—aj sin(27rA^) (1.1) j=i is the most famous example of a continuous but nowhere differentiable function. Weierstrass proved that this function is nowhere differentiable for some of these values of A and a, while Hardy [6] gave the first proof for all such A and a. The dimension of the graph ofthe function is suggested to be 2 — a. The box dimension was proved to be the case (under some mild condition)[l], but it remains open 5 The Fractal dimension of the Weierstrass type functions 6 whether the Hausdorff dimension of the graph to have the same value. In Chapter 2 Mass distribution principle [1] is used to show that the Hausdorff dimension of the graph of the Weierstrass function is very closed to 2 — a where A is large[5 . This technique is also used in chapter 3 to estimate the Hausdorff dimension of the graph of Tagaki function[2], a class of resemble functions. In another consideration, we let H — [0,1]°° endowed with the uniform probability measure, and let 0 — {〜，Oi, • • .}• We introduce a parameter to the Weierstrass function oo fe{x) 二 E A—^_ sin(27rA^ + Oj). 3 = l By showing that the t-energy of fjLe, /tM = / / (Or-")2 + (/,(�— /,("))2)t/2 is finite for t < 2 — a, Hunt[7] concludes that the Hausdorff dimension of the graph of fe over [0，1] is 2 — a for almost every 0 G H dues to the potential theoretic method where /i^ is induced by Lebesgue measure C on [0,1 . Self-affine sets form another important class of fractal sets. An affine map S : R^ ~^ EP is a transformation of the form S{x) = T{x) + b where T is a linear transformation on R" (representable by an n x n matrix) and b is a vector in R^. Suppose each Si {i = 1，...，m) is a contracting affine contractions. Then by a theorem of Hutchinson [8] there is a unique compact invariant set F with the property that F = USi{F). We call F a self-affine set. Like the Weierstrass function, the Hausdorff dimension of the self-affine set is difficult to obtain. Potential theoretic method[l] is used to show that for almost The Fractal dimension of the Weierstrass type functions 7 all bi there is a formula d{Ti, •.. ’ T^) expressing the HausdorfF dimension where d{Ti, • •., Tm) is a function depended on the singular value of T] [3 . Then, self-affine sets in the plane can also be considered as curves of func- tions by suitable choice of affine transformations. Given a collection of points (xo, Vo), • • •，{xm, ym)- Let Si be the affine transformation represented in matrix notation with respect to {x, y) coordinates by (X \ ( 1/m 0 \ ( X \ ( {i — l)/m� Si — + • \ y ) \ «i Ci } \ y J \ bi The properties Si{{xo,yo)) = {xi,yi) and Si[{xm,ym)) = (^m+i,Wi) are hold to ensure that the self-affine set is a curve. In the thesis we discuss the cases that the self-affine curves are differentiable or fractal type. The box dimension for the fractal type is l + log(CiH hc^)/logm[9]. Falconer[2] proves that the formula d(Ti, • • • ,Tm) has the same value in this case. Also the self-affine curve of the difFerentable case is studied on some special condition. Finally, recurrent set[ll] is introduced which is primarily based on an inter- play between algebra and analysis. The recurrent sets are the limits of a sequence of compact subsets of R^ associated with iterates of a given word under an ap- propriate free semigroup endomorphism. Its description is a powerful method of describing and generating space-filling curves, graphs of nowhere differentiable continuous functions. Chapter 2 Preliminaries 2.1 Box dimension and Hausdorff dimension Box dimension is one of the most widely used dimensions. It is easy to cal- culate and is widely used by the other scientists, even thought it lacks many desirable mathematical properties. For a non-empty bounded subset E, Let E\ 二 sup{|o; — y\ : X, y G E} be the diameter of a set E C M^. Let Nr{E) be the smallest number of sets of diameter r that can cover E. The lower and upper box dimensions of E are defined as dimB^ = liminflog， r—O — log r 厂 ^ .. logNr(E) dim^jC/ = lim sup r^^0 - log r respectively. If these two expressions are equal, we refer to the common value as the box dimension of E, i.e., .^ 1. logjVrQg) dimsE 二 lim ^^. r^0 — logr Thus the least number of sets of diameter r which can cover E is roughly of order r_s where s 二 dims^. The box dimension is unchanged if the covering set of E is replaced by closed balls, cubes, r-mesh cubes. 8 The Fractal dimension of the Weierstrass type functions 9 On the other hand, the Hausdorff dimension is defined in terms of a measure and is more sophisticated than the box dimension. Suppose that F is a subset of M" and s is a non-negative number. For any 6 > 0, we define ‘oo 飞 n's{F) = inf Y1 \Ui\' : {Ui} is a ^-cover of F \ (2.1) .i=l J As S decreases, the class of permissible covers of F in (2.1) is reduced. Therefore, the infimum %l{F) increases and so approaches a limit as 5 ~> 0.