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Bifurcation in Complex Quadrat Ic Polynomial and Some Folk Theorems Involving the Geometry of Bulbs of the Mandelbrot Set

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Bifurcation in Complex Quadrat Ic Polynomial and Some Folk Theorems Involving the Geometry of Bulbs of the Mandelbrot Set Bifurcation in Complex Quadrat ic Polynomial and Some Folk Theorems Involving the Geometry of Bulbs of the Mandelbrot Set Monzu Ara Begum A Thesis in The Department of Mat hemat ics and St at ist ics Presented in Partial F'uffilment of the Requirements for the Degree of Master of Science at Concordia University Montreal, Quebec, Canada May 2001 @Monzu Ara Begum, 2001 muisitions and Acquisitions et BiMiiriic Services senrices bibliographiques The author has granted a non- L'auteur a accordé une licence non exclusive licence aiiowing the exclusive permettant à la National Li'brary of Caoada to Bibliothèque nationale du Canada de reproduce, loan, distribute or seiî reproduire, prêtex' distribuer ou copies of this thesis in microform, vendre des copies de cette thèse sous paper or electronic formats. la forme de micmfiche/nlm. de reproduction sur papier ou sur format électronique. The auîhor retains ownership of the L'auteur conserve la propriété du copyright in this thesis. Neither the droit d'auteur qui protège cette thèse. thesis nor substantial extracts fiom it Ni La îhèse ni des extraits substantiels may be printed or otherwise de celle-ci ne doivent être imprimés reproduced without the author's ou autrement reproduits sans son permission. autorisation. Abstract Bifurcation in Complex Quadratic Polynomial and Some Folk Theorems Involving the Geometry of Bulbs of the Mandelbrot Set Monzu Ara Begum The Mandelbrot set M is a subset of the parameter plane for iteration of the complex quadratic polynomial Q,(z) = z2 +c. M consists of those c values for which the orbit of O is bounded. This set features a basic cardioid shape from which hang numerous 'bulbs' or 'decorations'. Each of these bulbs is a large disk that is directly attached to the main cardioid together with numerous other smaller bulbs and a prominent 'antenna'. In this thesis we study the geometry of bulbs and some 'folk theorems' about the geometry of bulbs involving spokes of the antenna. Acknowledgment s 1 would like to express my gratitude to my supervisor, professor Pawel G6ra of Con- cordia University, for his guidance of this thesis. Without his help and valuable suggestion it was impossible to mite this thesis. 1 am also grateful to the Depart- ment of Mathematics and Statistics of Concordia University for financial support during my Master's program. Contents 1 Introduction 1 1.1 General Introduction ........................... 1 1.2 Scopeofthethesis ............................ 2 2 Preliminaries 3 2.1 Fixed points. periodic points and their nature ............. 3 2.2 Fatou sets and Julia sets ......................... 5 2.3 Some properties of the Julia and the Fatou sets ............ 9 3 The Mandelbrot Set and Bifurcation in Complex Quadratic Polyno- mial 14 3.1 Complex Quadratic polynomial. Filled Julia set and the Mandelbrot set 15 3.2 Role of the critical orbit ......................... 19 3.3 Bifurcation in Q, (z) and visible bulbs of the Mandelbrot set ..... 19 3.4 Algorithm for the Mandelbrot set .................... 25 3.5 Periods of the bulb ............................ 26 3.6 Rotation numbers ............................. 29 4 Geometry of bulbs in the Mandelbrot set. Farey tree and Fibonacci sequence 32 4.1 Introduction ................................ 32 4.2 Farey addition. Farey tree. Fibonacci sequence ............. 33 4.3 Angle doubling mod 1 .......................... 39 4.4 Itinerary and Binary Expansion: .................... 42 4.5 External Rays ............................... 44 4.6 Idea of Limbs ............................... 47 4.7 Rays landing on the p/q bulb ...................... 47 4.8 An algorithm for computing the angles of rays landing at c(p/q).... 50 4.9 The size of Limbs and the Farey Tree .................. 54 4.10 The Fibonacci sequence ......................... 58 4.1 1 Conclusion ................................. 60 Bibliography 61 Chapter 1 Introduction 1.1 General Introduction In the study of complex dynarnical systerns the Mandelbrot set is one of the most intricate and beautiful objects in Mathematics. The geometry in the Mandelbrot set is quite complicated. In the last fifteen years many mathematicians such as Paul Blanchard [BL], Pau Atela [A],Bodil Branner [BR],R.L. Devaney [D2],E.Lau and D. Sclileicher Yoccoz [LS]and recently R.L.Devaney [Dl] and R.L.Devaney and M-Morenno-Rocha [DM] studied the geometry of the bulbs and antennae in the Mandelbrot set. In this thesis we study the Mandelbrot set and geometry of bulbs. We follow [Dl], [DM]and [02]. The study of the geometry of the Mandelbrot set needs basic concepts on iteration theory, Fatou set and Julia sets. First we review these basic concepts. Our main work is on 'some folk theorems' about the geometry Figure 1.1 - 1.2 Scope of the thesis In Chapter 2 we revien- some basic concepts of Discrete Dynamical systerns, CompIes Analysis, Fatou and Julia sets for a cornplex polynomial. In Chapter 3 the definition and construction of the Mandelbrot set and some prop- erties of the Mandelbrot set are given. We aIso discuss period doubling bifurcation. We present our niain work on the geometry of bulbs in Chapter 4. In this Chapter Ive discuss the geomerry of the bulbs, geometry of the aiitennas and esternal rays. The main tool of the discussior~is the doubling fiiiictioii aiid rotation map. We state md prove some folk theorcms in this Chapter. Chapter 2 Preliminaries In this chapter we review some basic concepts of complex analysis, discrete dynamical systems. Throughout this chapter we follow [BE],[D3] and [KF]. 2.1 Fixed points, periodic points and their nature Definition 2.1 Let @, = @ U {m) and P be a polynomzal of degree n 3 1. A point zo E 2s sazd to be a fized point of P if P(ro)= zo. Moreover, zo 2s (2) attracting zf 1 Pf(zo)l< 1; (ii) super-attractzng 2j Pf(ro)= 0; (222) repellzng if 1 Pt(ro ) 1 > 1 ; (zv) neutml if 1 P1(zo)1 = 1; Definition 2.2 Let P be a polynomial of degree n > I and a E @,. The sequence is called the orbit of zo under P. The point zo is a peBodic point of period k if Pk(zo)= zo and Pq(zo) # zo for 1 5 q < k. If zo is a periodic point of period k, then the sequence {zo,P(zo), P2(zo), P3(4.. Pk-l(~o)) is a k-cgcle. Lemma 2.1 A point zo is an attractzng fied point of a polynomial P if and only if 3 a nezghborhood U of zo such that (i) P(U) C U; Definition 2.3 The basin of attraction ofzo 2s the set of al1 points whose orbit tends to q-, and it is denoted by B(zo),that is B(zO)= {Z : Pn(z)+ 20, as n + m). Example 2.1 Let P : & + @, be defined by P(z)= z2. Let U = {z : Izl < r < 1) be a neighborhood of z = O. Then Inductively, we get Pn(U)= {z : [zl 5 rZn). SO Pn(U)= {O). Definition 2.4 Suppose zo is a periodic point of period k for the polpomial P. Let A = (Pk)'(zO)Then the k cycle {zo,P(zo), P2(zo), P3(z0)? ..., Ph-'(z0)}is 4 (2) attrncting if IXI < 1; (iz) super-attractzng if X = 0: (iii) repellzng zj 1 XI > 1; (zv) neutral if IXI = 1; 2.2 Fatou sets and Julia sets To study the Mandelbrot set we need the concept of Julia set. The Julia set is related to the Fatou set. In this subsection we define Fatou and Julia sets of a polynomial P in terms of equicontinuity of the family {Pn),>o.- Since equicontinuity is closely related to normal families of analytic functions, using the notion of normality we can get more information about the Fatou and Julia sets. Hence, we discuss normality. Definition 2.5 Let (X,d) and (Xi, dl) be two rnetric spaces. A family 7 of maps of (X,d) into (Xi, dl) is equicontinuous ut xo if and only if for every E > O there exists 6 > O such that for al1 x E X and for al1 f E 3 The family 3 is eqzGicontinuous on a subset Xo of X if it is equicontinuous at each point xo of Xo. Theorem 2.1 [BE]Let 3 be any family of maps , each mapping (X, d) into (Xi,di). Then there is a maximal open subset of X on which 3 is equicontinuous. In particular, 5 if P maps a metric space (X,d) into itself, then there as a maximal open subset of A' on which the family of iterates Pn zs equicontinous. Definition 2.6 Let P be a complex polynomzal of degree n 2 2. Then the Fatou set F(P) of P is the maximal open subset of @, on which {Pn),>o- is equicontinous and the Julza set J(P) of P is the complement of the Fatou set in &. In other words, Fatou set is the stable set and Julia set is the unstable set. We see the Fatou and the Julia sets in the following example. Definition 2.7 A family 3 of maps from (Xi,di) into (X2,d2) is said to be a normal or a normal farnily in Xi if every infinite sequence of functions from F contains a subsequence which converges locally unijormly on every compact subset of ATi. Example 2.2 Let P : @, -+ @, be defined by P(z) = z2. Here fixed points of P are 0,l and oo. P'(z) = 22 and IP'(Z)~.=~= O. Thus O is a super-attracting fixed point of P. Also IP'(z) l,=' = 2 > 1 and thus 1 is a repelling fixed point of P. If we consider z E @, such that lzl < 1, Le., the open unit disc.
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