DOCSLIB.ORG

Generalizations of Douady's Magic

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Generalizations of Douady's Magic GENERALIZATIONS OF DOUADY'S MAGIC FORMULA ADAM EPSTEIN AND GIULIO TIOZZO Abstract. We generalize a combinatorial formula of Douady from the main car- dioid to other hyperbolic components H of the Mandelbrot set, constructing an explicit piecewise linear map which sends the set of angles of external rays landing on H to the set of angles of external rays landing on the real axis. 2 In the study of the dynamics of the quadratic family fc(z) := z + c, c 2 C, a central object of interest is the Mandelbrot set (n) M := fc 2 C :(fc (0))n≥0 is boundedg: The Mandelbrot set contains two notable smooth curves: namely, the intersection with the real axis M\R = [−2; 1=4], and the main cardioid of M, which is the set of parameters c for which fc has an attracting or indifferent fixed point in c (of course, this is smooth except at the cusp c = 1=4). Let R denote the set of angles of external rays landing on the real axis, and Ξ the set of angles of external rays landing on the main cardioid. The following \magic formula" is due to Douady (for a proof see [Bl]): Theorem 1 (Douady). The map 1 θ 1 2 + 4 if 0 ≤ θ < 2 T (θ) := 1 θ 1 4 + 4 if 2 < θ ≤ 1 sends Ξ into R. In this note, we prove the following generalization of Theorem 1 to other hyperbolic components. Let D(θ) := 2θ mod 1 denote the doubling map. Theorem 2. Let H be a hyperbolic component which belongs to a vein V of the Mandelbrot set, not in the 1=2-limb. Let AH , BH be the binary expansions of the root of H, and let δV be the complexity of V . Then the map δV ΦH (θ) := D (BH AH · θ) sends the set of external angles of rays landing on the upper part of H into the set R of angles of rays landing on the real axis. To illustrate such a formula, let us consider the \kokopelli" component of period 4, which has root of angles θ1 = 3=15 = :0011, θ2 = 4=15 = :0100, thus AH = 0011, University of Warwick, UK, [email protected]. University of Toronto, Canada, [email protected]. 1 2 ADAM EPSTEIN AND GIULIO TIOZZO Figure 1. The magic formula for the kokopelli component. The angle θ = 1=5 which lands at the root of its component is sent to ΦH (θ) = 21=40 which lands on the real axis. BH = 0100. This component lies on the principal vein with tip θ = 1=4, hence its complexity is δV = 1. Thus, 67 θ Φ (θ) := 1000011 · θ = + : H 128 128 Let us note that if H is the main cardioid, then δV = 0, AH = 0, BH = 1, hence we recover Douady's original formula from Theorem 1. Remarks. Note that the map ΦH is very far from being surjective: indeed, the set of angles of rays landing on a hyperbolic component has Hausdorff dimension zero, while the set R has dimension 1. During the preparation of this paper we have been informed of the generalization of Douady's formula by Bl´eand Cabrera [BC]. Let us remark that our formula is quite different, as the one in [BC] does not map into the real axis but rather into the GENERALIZATIONS OF DOUADY'S MAGIC FORMULA 3 0.70 0.65 0.60 0.55 0.50 0.45 0.0 0.1 0.2 0.3 0.4 0.5 Figure 2. The graph of the function Ψ(x) discussed in Remark 4. The red dots correspond to points on the graph of Douady's magic formula function T = 'H for the main cardioid, the (thick) blue dots to the image of 'H for the rabbit component, and the green dots to the kokopelli component. tuned copies of the real axis inside small Mandelbrot sets. In fact, we will describe their formula and how it relates to ours in Section 1.2. Acknowledgements. We thank S. Koch and C. T. McMullen for useful conversa- tions. G.T. is partially supported by NSERC and the Alfred P. Sloan Foundation. 1. Combinatorics of external rays 1.1. Real rays and the original formula. Consider the Riemann map ΦM : 0 C n D ! CnM, which is unique if we normalize it so that ΦM (1) = 1,ΦM (1) = 1. For each θ 2 R=Z, we define the external ray of angle θ as 2πiθ RM (θ) := fΦM (re ): r > 1g: 2πiθ The ray RM (θ) lands if there exists γ(θ) := limr!1+ ΦM (re ). It is conjectured that the ray RM (θ) lands for any θ 2 R=Z; to circumvent this issue, we define as RbM (θ) the impression of RM (θ), and the set of real angles as Rb := fθ 2 R=Z : RbM (θ) \ R 6= ;g: Conjecturally, this is exactly the set of angles for which the corresponding ray lands on the real axis. For further details, see e.g. [Za]. 4 ADAM EPSTEIN AND GIULIO TIOZZO The following lemma gives a characterization of Rb in terms of the dynamics of the doubling map. Lemma 3 ([Do]). The set of real angles equals n Rb := fθ 2 R=Z : jD (θ) − 1=2j ≥ jθ − 1=2j 8n 2 Ng: 1 Proof of Theorem 1. By symmetry, we can assume 0 < θ < 2 . Since the external ray n of angle θ lands on the main cardioid, then the forward orbit P := (D (θ))n2N of θ 1 θ the angle θ does not intersect the half-circle I = ( 2 + 2 ; 2 ) (the one which contains 0) (see [BS]). The preimage of I by the doubling map is the union of two intervals θ θ 1 θ 1 θ (− 4 ; 4 ) [ ( 2 − 4 ; 2 + 4 ). In particular, the forward orbit of θ does not intersect 1 θ 1 θ 0 0 θ J = ( 2 − 4 ; 2 + 4 ). Thus, if we look at the orbit of θ = T (θ), we have 2θ = 2 2= J and Dn(θ0) 2 P for any n ≥ 2, hence the forward orbit of θ0 does not intersect J, 0 hence θ is a real angle by Lemma 3. Remark 4. By Lemma 3, one recognizes that a way to map any angle θ 2 R=Z to the set of real angles is to consider the function 1 k 1 Ψ(x) := + inf D (x) − ; 2 k≥0 2 which indeed satisfies Ψ(R=Z) ⊆ Rb. As you can see from Figure 2, this function is discontinuous, while its graph \contains" the graphs of all the magic formula functions 'H given by Theorem 2 (which are indeed continuous) for all components H. 1.2. Tuning and the Bl´e-Cabreramagic formula. Given any hyperbolic com- ponent H in the Mandelbrot set, let us recall that there is a tuning map which sends the main cardioid to the hyperbolic component H, and the Mandelbrot set to a small copy of itself which contains H. In order to define the map precisely, let aH = :a1 : : : ak < bH = :b1 : : : bk denote the two external angles of rays landing at the root of H, and denote AH = a1 : : : ak and BH = b1 : : : bk the two corresponding finite binary words. Moreover, denote 0 0 aH = :a1 : : : akb1 : : : bk and bH = :b1 : : : bka1 : : : ak. The tuning map TH on the set of external angles is now defined as follows. If θ = .12 ::: is the binary expansion of θ, then the angle TH (θ) has binary expansion TH (θ) = :A1 A2 ::: where A0 = AH , A1 = BH . Then, the set ΞH := TH (Ξ) is the set of rays landing on the boundary of H. If θ 2 T = R=Z has infinite binary expansion (i.e. it is not a dyadic number) and S = s1 : : : sn is a finite word on the alphabet f0; 1g, we denote by S · θ the element of T n X −k −n S · θ := sk2 + 2 θ k=1 i.e. the point whose binary expansion is the concatenation of S and the binary expansion of θ. Recall that tuning behaves well with respect to concatenation, namely GENERALIZATIONS OF DOUADY'S MAGIC FORMULA 5 for any S; H and θ we have TH (S · θ) = TH (S) · TH (θ): We define the small real vein associated to H, and denote it as RH , to be the real vein of the small Mandelbrot set associated to H. Let us denote as RH the set of external angles of rays landing on RH , and as RbH the set of external angles of rays whose impression intersects RH . Clearly, RH0 = R if H0 is the main cardioid, and RH0 = R. −1 Proposition 5 (Bl´e-Cabrera[BC]). The map TH := TH ◦ T ◦ TH maps ΞH into the small real vein RbH . Moreover, this map (restricted to ΞH ) is piecewise affine: in fact, it can be written in terms of binary expansions as TH (θ) = BH AH · θ 0 if θ 2 (aH ; aH ), and TH (θ) = AH BH · θ 0 if θ 2 (bH ; bH ). Proof. By construction, the tuning map TH :Ξ n f0g ! ΞH n faH ; bH g is a bijection, so the first statement follows by looking at the diagram: T Ξ / Rb TH TH TH ΞH / RbH : Moreover, since T (θ) = 10 · θ for θ < 1=2 and tuning behaves nicely with respect to concatenation, we have −1 TH (θ) = TH (10 · TH (θ)) = TH (10) · θ which proves the second claim.
Load more

Recommended publications
  • On the Combinatorics of External Rays in the Dynamics of the Complex Henon
    ON THE COMBINATORICS OF EXTERNAL RAYS IN THE DYNAMICS OF THE COMPLEX HENON MAP. A Dissertation Presented to the Faculty of the Graduate School of Cornell University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy by Ricardo Antonio Oliva May 1998 c Ricardo Antonio Oliva 1998 ALL RIGHTS RESERVED Addendum. This is a slightly revised version of my doctoral thesis: some typing and spelling mistakes have been corrected and a few sentences have been re-worded for better legibility (particularly in section 4.3). Also, to create a nicer pdf document with hyperref, the title of section 3.3.2 has been made shorter. The original title was A model for a map with an attracting fixed point as well as a period-3 sink: the (3-1)-graph. ON THE COMBINATORICS OF EXTERNAL RAYS IN THE DYNAMICS OF THE COMPLEX HENON MAP. Ricardo Antonio Oliva , Ph.D. Cornell University 1998 We present combinatorial models that describe quotients of the solenoid arising from the dynamics of the complex H´enon map 2 2 2 fa,c : C → C , (x, y) → (x + c − ay, x). These models encode identifications of external rays for specific mappings in the H´enon family. We investigate the structure of a region of parameter space in R2 empirically, using computational tools we developed for this study. We give a combi- natorial description of bifurcations arising from changes in the set of identifications of external rays. Our techniques enable us to detect, predict, and locate bifurca- tion curves in parameter space. We describe a specific family of bifurcations in a region of real parameter space for which the mappings were expected to have sim- ple dynamics.
    [Show full text]
  • Infinitely Renormalizable Quadratic Polynomials 1
    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 352, Number 11, Pages 5077{5091 S 0002-9947(00)02514-9 Article electronically published on July 12, 2000 INFINITELY RENORMALIZABLE QUADRATIC POLYNOMIALS YUNPING JIANG Abstract. We prove that the Julia set of a quadratic polynomial which ad- mits an infinite sequence of unbranched, simple renormalizations with complex bounds is locally connected. The method in this study is three-dimensional puzzles. 1. Introduction Let P (z)=z2 + c be a quadratic polynomial where z is a complex variable and c is a complex parameter. The filled-in Julia set K of P is, by definition, the set of points z which remain bounded under iterations of P .TheJulia set J of P is the boundary of K. A central problem in the study of the dynamical system generated by P is to understand the topology of a Julia set J, in particular, the local connectivity for a connected Julia set. A connected set J in the complex plane is said to be locally connected if for every point p in J and every neighborhood U of p there is a neighborhood V ⊆ U such that V \ J is connected. We first give the definition of renormalizability. A quadratic-like map F : U ! V is a holomorphic, proper, degree two branched cover map, whereT U and V are ⊂ 1 −n two domains isomorphic to a disc and U V .ThenKF = n=0 F (U)and JF = @KF are the filled-in Julia set and the Julia set of F , respectively. We only consider those quadratic-like maps whose Julia sets are connected.
    [Show full text]
  • Geography of the Cubic Connectedness Locus : Intertwining Surgery
    ANNALES SCIENTIFIQUES DE L’É.N.S. ADAM EPSTEIN MICHAEL YAMPOLSKY Geography of the cubic connectedness locus : intertwining surgery Annales scientifiques de l’É.N.S. 4e série, tome 32, no 2 (1999), p. 151-185 <http://www.numdam.org/item?id=ASENS_1999_4_32_2_151_0> © Gauthier-Villars (Éditions scientifiques et médicales Elsevier), 1999, tous droits réservés. L’accès aux archives de la revue « Annales scientifiques de l’É.N.S. » (http://www. elsevier.com/locate/ansens) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systé- matique est constitutive d’une infraction pénale. Toute copie ou impression de ce fi- chier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Ann. sclent. EC. Norm. Sup., 4e serie, t 32, 1999, p. 151 a 185. GEOGRAPHY OF THE CUBIC CONNECTEDNESS LOCUS: INTERTWINING SURGERY BY ADAM EPSTEIN AND MICHAEL YAMPOLSKY ABSTRACT. - We exhibit products of Mandelbrot sets in the two-dimensional complex parameter space of cubic polynomials. Cubic polynomials in such a product may be renormalized to produce a pair of quadratic maps. The inverse construction intertwining two quadratics is realized by means of quasiconformal surgery. The associated asymptotic geography of the cubic connectedness locus is discussed in the Appendix. © Elsevier, Paris RESUME. - Nous trouvons des produits de Fensemble de Mandelbrot dans 1'espace a deux variables complexes des polynomes cubiques. La renormalisation d'un polyn6me cubique appartenant a un tel produit donne deux polyn6mes quadratiques. Le precede inverse qui entrelace deux polynomes quadratiques est obtenu par chirurgie quasiconforme.
    [Show full text]
  • The External Boundary of the Bifurcation Locus M2 1
    THE EXTERNAL BOUNDARY OF THE BIFURCATION LOCUS M2 V. TIMORIN Abstract. Consider a quadratic rational self-map of the Riemann sphere such that one critical point is periodic of period 2, and the other critical point lies on the boundary of its immediate basin of attraction. We will give explicit topological models for all such maps. We also discuss the corresponding parameter picture. 1. Introduction 1.1. The family V2. Consider the set V2 of holomorphic conjugacy classes of qua- dratic rational maps that have a super-attracting periodic cycle of period 2 (we follow the notation of Mary Rees). The complement in V2 to the class of the single map z 1/z2 is denoted by V . The set V is parameterized by a single com- 7→ 2,0 2,0 plex number. Indeed, for any map f in V2,0, the critical point of period two can be mapped to , its f-image to 0, and the other critical point to 1. Then we obtain a map of the∞ form − a f (z)= , a = 0 a z2 + 2z 6 holomorphically conjugate to f. Thus the set V is identified with C 0. 2,0 − The family V2 is just the second term in the sequence V1, V2, V3,... , where, by def- inition, Vn consists of holomorphic conjugacy classes of quadratic rational maps with a periodic critical orbit of period n. Such maps have one “free” critical point, hence each family Vn has complex dimension 1. Note that V1 is the family of quadratic polynomials, i.e., holomorphic endomorphisms of the Riemann sphere of degree 2 with a fixed critical point at .
    [Show full text]
  • 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.
    [Show full text]
  • Topological Entropy of Quadratic Polynomials and Dimension of Sections of the Mandelbrot Set
    TOPOLOGICAL ENTROPY OF QUADRATIC POLYNOMIALS AND DIMENSION OF SECTIONS OF THE MANDELBROT SET GIULIO TIOZZO Abstract. A fundamental theme in holomorphic dynamics is that the local geometry of parameter space (e.g. the Mandelbrot set) near a parameter re- flects the geometry of the Julia set, hence ultimately the dynamical properties, of the corresponding dynamical system. We establish a new instance of this phenomenon in terms of entropy. Indeed, we prove an \entropy formula" relating the entropy of a polynomial restricted to its Hubbard tree to the Hausdorff dimension of the set of rays landing on the corresponding vein in the Mandelbrot set. The results con- tribute to the recent program of W. Thurston of understanding the geometry of the Mandelbrot set via the core entropy. 1. Introduction Let us consider the family of quadratic polynomials 2 fc(z) := z + c; with c 2 C: The filled Julia set K(fc) of a quadratic polynomial fc is the set of points which do not escape to infinity under iteration, and the Julia set J(fc) is the boundary of K(fc). The Mandelbrot set M is the connectedness locus of the quadratic family, i.e. M := fc 2 C : the Julia set of fc is connectedg: A fundamental theme in the study of parameter spaces in holomorphic dynamics is that the local geometry of the Mandelbrot set near a parameter c reflects the geometry of the Julia set J(fc), hence it is related to dynamical properties of fc. In this paper we will establish a new instance of this principle, by comparing the topological entropy of quadratic polynomials to the Hausdorff dimension of certain sets of external rays.
    [Show full text]
  • Arxiv:2009.02788V1 [Math.DS] 6 Sep 2020 by 0 the Set of Angles for Which Θ Or One of Θ Lands at 0
    PERIODIC POINTS AND SMOOTH RAYS CARSTEN LUNDE PETERSEN AND SAEED ZAKERI Abstract. Let P : C ! C be a polynomial map with disconnected filled Julia set KP and let z0 be a repelling or parabolic periodic point of P . We show that if the connected component of KP containing z0 is non-degenerate, then z0 is the landing point of at least one smooth external ray. The statement is optimal in the sense that all but one ray landing at z0 may be broken. 1. Introduction It has been known since the pioneering work of Douady and Hubbard in the early 1980’s that a repelling or parabolic periodic point of a polynomial map with connected Julia set is the landing point of finitely many external rays of a common period and combinatorial rotation number [H, M, P]. When suitably formulated, this statement remains true for polynomials with disconnected Julia set although in this case one must also allow broken external rays that crash into the escaping (pre)critical points [LP]. To state this precisely, let P : C ! C be a polynomial of degree D ≥ 2 with disconnected filled Julia set KP . The external rays of P can be defined in terms of the gradient flow of the Green’s function G of P in C r KP . They consist of smooth field lines that descend from 1 and approach KP , as well as their limits which are broken rays that abruptly turn when they crash into a critical point of G . For each ± θ 2 T := R/Z the polynomial P has either a smooth ray Rθ or a pair Rθ of broken + − rays which descend from 1 at the normalized angle θ.
    [Show full text]
  • VMD User's Guide
    VMD User’s Guide Version 1.9.3 November 27, 2016 NIH Biomedical Research Center for Macromolecular Modeling and Bioinformatics Theoretical and Computational Biophysics Group1 Beckman Institute for Advanced Science and Technology University of Illinois at Urbana-Champaign 405 N. Mathews Urbana, IL 61801 http://www.ks.uiuc.edu/Research/vmd/ Description The VMD User’s Guide describes how to run and use the molecular visualization and analysis program VMD. This guide documents the user interfaces displaying and grapically manipulating molecules, and describes how to use the scripting interfaces for analysis and to customize the be- havior of VMD. VMD development is supported by the National Institutes of Health grant numbers NIH 9P41GM104601 and 5R01GM098243-02. 1http://www.ks.uiuc.edu/ Contents 1 Introduction 10 1.1 Contactingtheauthors. ....... 11 1.2 RegisteringVMD.................................. 11 1.3 CitationReference ............................... ...... 11 1.4 Acknowledgments................................. ..... 12 1.5 Copyright and Disclaimer Notices . .......... 12 1.6 For information on our other software . .......... 14 2 Hardware and Software Requirements 16 2.1 Basic Hardware and Software Requirements . ........... 16 2.2 Multi-core CPUs and GPU Acceleration . ......... 16 2.3 Parallel Computing on Clusters and Supercomputers . .............. 17 3 Tutorials 18 3.1 RapidIntroductiontoVMD. ...... 18 3.2 Viewing a molecule: Myoglobin . ........ 18 3.3 RenderinganImage ................................ 20 3.4 AQuickAnimation................................. 20 3.5 An Introduction to Atom Selection . ......... 21 3.6 ComparingTwoStructures . ...... 21 3.7 SomeNiceRepresenations . ....... 22 3.8 Savingyourwork.................................. 23 3.9 Tracking Script Command Versions of the GUI Actions . ............ 23 4 Loading A Molecule 25 4.1 Notes on common molecular file formats . ......... 25 4.2 Whathappenswhenafileisloaded? . ....... 26 4.3 Babelinterface .................................
    [Show full text]
  • Drawing and Computing External Rays in the Multiple- Spiral Medallions of the Mandelbrot Set
    Drawing and computing external rays in the multiple- spiral medallions of the Mandelbrot set M. Romera *, G. Alvarez, D. Arroyo, A.B. Orue, V. Fernandez, G. Pastor Instituto de Física Aplicada, Consejo Superior de Investigaciones Científicas, Serrano 144, 28006 Madrid, Spain ___________________________________________________________________________ Abstract The multiple-spiral medallions are beautiful decorations of the Mandelbrot set. Computer graphics provide an invaluable tool to study the structure of these decorations with central symmetry, formed by an infinity of baby Mandelbrot sets that have high periods. Up to now, the external arguments of the external rays landing at the cusps of the cardioids of these baby Mandelbrot sets could not be calculated. Recently, a new algorithm has been proposed in order to calculate the external arguments in the Mandelbrot set. In this paper we use an extension of this algorithm for the calculation of the binary expansions of the external arguments of the baby Mandelbrot sets in the multiple-spiral medallions. PACS: 05.45.Df Keywords: Mandelbrot set; External arguments; Binary expansions ___________________________________________________________________________ 1. Introduction As is well known, the Mandelbrot set [1] can be defined by M =∈C k →∞/ →∞ k {c: fc (0) as k }, where fc (0) is the k-iteration of the parameter- =2 + = dependent quadratic function fzc ( ) z c ( z and c complex) for the initial value z0 0 (the critical point). Douady and Hubbard studied the Mandelbrot set by means of the external arguments theory [2-4] that was popularized as follows [5]. Imagine a capacitor made of an aluminum bar, shaped in such a way that its cross-section is M , placed along the axis of a hollow metallic cylinder (Fig.
    [Show full text]
  • A GENERALIZATION of DOUADY's FORMULA Gamaliel Blé Carlos Cabrera (Communicated by Enrique Pujals) 1. Introduction. in This Pa
    DISCRETE AND CONTINUOUS doi:10.3934/dcds.2017267 DYNAMICAL SYSTEMS Volume 37, Number 12, December 2017 pp. 6183{6188 A GENERALIZATION OF DOUADY'S FORMULA Gamaliel Ble´∗ Divisi´onAcad´emicade Ciencias B´asicas,UJAT Km. 1, Carretera Cunduac´an{Jalpade M´endez C.P. 86690, Cunduac´anTabasco, M´exico Carlos Cabrera Instituto de Matem´aticasde la UNAM, Unidad Cuernavaca Av. Universidad s/n. Col. Lomas de Chamilpa C.P. 62210, Cuernavaca, Morelos, M´exico (Communicated by Enrique Pujals) Abstract. The Douady's formula was defined for the external argument on the boundary points of the main hyperbolic component W0 of the Mandelbrot set M and it is given by the map T (θ) = 1=2 + θ=4. We extend this formula to the boundary of all hyperbolic components of M and we give a characterization of the parameter in M with these external arguments. 1. Introduction. In this paper we will restrict our attention to the dynamics of 2 complex quadratic polynomials Pc(z) = z +c: In this case, the dynamical plane can be decomposed into two complementary sets: The filled Julia set Kc which consists of all points with bounded orbit and the basin of infinity Ac(1). The boundary of Kc is called the Julia set Jc. ^ When we consider the polynomial Pc in the extended complex plane C = C [ f1g, the point at infinity is a super-attracting fixed point and hence there exists a neighborhood of infinity Uc and an analytic isomorphism, known as the B¨ottcher 0 map, φc : Uc ! fz 2 C : jzj > Rg; such that φc(1) = 1; φc(1) = 1 and 2 φc ◦Pc = [φc] .
    [Show full text]
  • ON BIACCESSIBLE POINTS of the MANDELBROT SET 1. Introduction
    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 134, Number 8, Pages 2239–2250 S 0002-9939(06)08559-5 Article electronically published on March 14, 2006 ON BIACCESSIBLE POINTS OF THE MANDELBROT SET SAEED ZAKERI (Communicated by Linda Keen) Abstract. This paper provides a description for the quadratic polynomials on the boundary of the Mandelbrot set M which are typical in the sense of harmonic measure. In particular, it is shown that a typical point on the boundary of M has a unique parameter ray landing on it. Applications of this result in the study of embedded arcs in M and the lamination associated with M are given. 1. Introduction 2 Consider the complex quadratic family {fc : z → z +c}c∈C and its connectedness locus M, commonly known as the “Mandelbrot set.” This paper will give a short proof of the following Theorem 1. There exists a Borel set X ⊂ ∂M of full harmonic measure such that for every c ∈ X: (i) the Julia set Jc = J(fc) is locally connected and full; { ◦2 } (ii) the post-critical orbit c, fc(c),fc (c),... is evenly distributed with respect to the harmonic measure on Jc (in particular it is dense in Jc); (iii) M is locally connected at c, only one parameter ray lands at c,theim- pression of this ray is trivial, and no other parameter ray contains c in its impression. Moreover, the dynamic ray of the same angle lands at the critical value c ∈ Jc. The statement can be recovered from the work of Bruin and Schleicher [BSc] in which they use symbolic dynamics to study various combinatorial and topological aspects of the quadratic family.
    [Show full text]
  • Algorithms for Computing Angles in the Mandelbrot
    ALGORITHMS FOR COMPUTING ANGLES IN THE MANDELBROT SET A. Douady 48 rue Monsieur le Prince 74006 Paris FRANCE 1. NOTATIONS 2 f ! ZM +C, Z€(E. fn = f iterated n times.. K = {z|fn(z) fi «>} the filled-in Julia set. c ' c J = 3K the Julia set c c f has two fixed points a(c) and 3(c). Convention: 3(c) is the most repulsive (the one on the right). M = {c|K is connected} = {c|0 e K }. 1 c ■ c VQ = {c|0 is periodic for f } (centers of hyperbolic com­ ponents) . V2 = {c|0 is strictly preperiodic} (Misiurewicz points). P, = {c|f has a rational neutral cycle} (roots of hyperbolic components). Chaotic Dynamics Copyright © 1986 by Academic Press, Inc. and Fractals 155 All rights of reproduction in any form reserved. 156 A. Douady POTENTIAL AND EXTERNAL ARGUMENTS The potential G created by K is given by . \0 if z e K + Gc(z) =lim^Log |f£(z)|= J ° 2 Log|z| + E —-ry Log|l + 2n+l—fn(z)2 else n The map z H (z) = lim(f n (z))l/2 ' is well defined for all z e (C - Kc if Kc is connected, only for Gc (z) > Gc (0) if Kc is a Cantor set. The external argument with respect to K is Argc (z) = Arg(<t>c (z)). (The unit for arguments is the whole turn, not the radian.) For z e J = 8K , one can define one value of Argc (z) for each wauy of access to z in (C - Kc .
    [Show full text]