DOCSLIB.ORG

Homeomorphisms on Edges of the Mandelbrot Set

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Homeomorphisms on Edges of the Mandelbrot Set Homeomorphisms on Edges of the Mandelbrot Set Von der Fakult¨atf¨urMathematik, Informatik und Naturwissenschaften der Rheinisch-Westf¨alischen Technischen Hochschule Aachen genehmigte Dissertation zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften vorgelegt von Diplom-Mathematiker Wolf Jung aus Gelsenkirchen-Buer Berichter: Universit¨atsprofessor Dr. Volker Enss Universit¨atsprofessor Dr. Gerhard Jank Universit¨atsprofessor Dr. Walter Bergweiler Tag der m¨undlichen Pr¨ufung:3.7.2002 Diese Dissertation ist auf den Internetseiten der Hochschulbibliothek online verf¨ugbar. The author’s present address is: Wolf Jung Institut f¨urReine und Angewandte Mathematik RWTH Aachen, D-52062 Aachen, Germany [email protected] http://www.iram.rwth-aachen.de/∼jung This Ph.D. thesis was accepted by the Faculty of Mathematics, Computer Science and Natural Sciences at the RWTH Aachen, and the day of the oral examination was July 3, 2002. The thesis is available as a pdf-file or ps-file from the author’s home page and from the RWTH-library http://www.bth.rwth-aachen.de/ediss/ediss.html. Hints for viewing the file: The pdf-version is recommended for reading on the screen. You should save it to your local disk and open it with Adobe’s Acrobat Reader (http://www.adobe.com), setting the display to maximal width (CTRL 3), which may require choosing single page mode. Clicking on the links on the current page will start your web browser. Clicking on references on other pages will send you approximately to the corresponding part of this manuscript, and you can return with the Reader’s back button. Program: All images have been produced with the DOS-program mandel.exe, which is available from the author’s home page. The algorithm used for draw- ing external rays will be described in [J2]. Although it is not considered to be part of this thesis, writing the program and researching on holomorphic dynamics have benefited from each other in turns. Acknowledgment I am most grateful to Volker Enss for his continuous interest and engaged support of every aspect of my work. I wish to thank Walter Bergweiler and Gerhard Jank for useful hints and for undertaking the labor of refereeing this thesis. Thanks to Johannes Riedl for proofreading early versions of the manuscript, many corrections and helpful suggestions, and for our most inspiring discussions on surgery and renormalization. Dierk Schleicher provided invaluable advice on the background in holomorphic dynamics, and his critical remarks influenced the course of my work, and motivated in particular the research for Section 8.1. I am grateful for useful hints from and inspiring discussions with Bodil Bran- ner, Xavier Buff, N´uriaFagella, Lukas Geyer, Peter Ha¨ıssinsky, Heinz Hanßmann, Karsten Keller, Hartje Kriete, Steffen Rohde, Rudolf Winkel and Tan Lei. I wish to thank all colleagues at the Institut f¨urReine und Angewandte Mathematik for the supporting and warm atmosphere. Part of this work was done on long evenings in some nice Caf´esof Aachen: Kittel, Labyrinth, Last Exit, Meisenfrei, Molkerei, Orient Expresso, Pontgarten, Wild Roses and Ohne Worte (Munich). 3 Contents Introduction 7 1 Summary of Results 11 1.1 Quasi-Conformal Surgery ............................ 11 1.2 The Homeomorphism h on an Edge ....................... 14 1.3 Comparison of Techniques and Results ..................... 16 1.4 Edges and Frames ................................ 18 1.5 Repelling Dynamics at Misiurewicz Points ................... 20 1.6 Combinatorial Surgery and Homeomorphism Groups ............. 22 1.7 Suggestions for Further Research ........................ 22 2 Background 25 2.1 Conformal Mappings ............................... 25 2.2 Quasi-Conformal Mappings ........................... 28 2.3 The Analytic Definition of Quasi-Conformal Mappings ............ 30 2.4 Extension by the Identity ............................ 31 2.5 Extension of Holomorphic Motions ....................... 32 2.6 Iteration of Rational Functions ......................... 33 3 The Mandelbrot Set 35 3.1 Iteration of Quadratic Polynomials ....................... 35 3.2 The Mandelbrot Set ............................... 40 3.3 Cycles and Hyperbolic Components ...................... 42 3.4 Correspondence of Landing Patterns ...................... 45 3.5 Limbs, Puzzles and Fibers ............................ 49 3.6 Combinatorial and Topological Models ..................... 51 3.7 Non-Hyperbolic Components .......................... 54 4 Renormalization and Surgery 56 4.1 Polynomial-Like Mappings ............................ 56 4.2 A Quasi-Regular Straightening Theorem .................... 59 4.3 Tuning ....................................... 63 4.4 Renormalization and Local Connectivity .................... 67 4 4.5 Examples of Homeomorphisms ......................... 71 5 Constructing Homeomorphisms 74 5.1 Combinatorial Definitions ............................ 74 5.2 Construction of gc ................................ 78 5.3 Properties of h .................................. 83 5.4 The Exterior of Kc , M and D .......................... 85 5.5 Bijectivity ..................................... 89 5.6 Continuity and Analyticity ........................... 91 6 Edges 96 6.1 Dynamic and Parameter Edges ......................... 96 6.2 Homeomorphisms on Edges ........................... 101 6.3 Graphs of Maximal Edges ............................ 102 7 Frames 106 7.1 Dynamic and Parameter Frames ........................ 106 7.2 Hierarchies of Homeomorphic Frames ...................... 109 7.3 The Structure of Frames ............................. 114 7.4 Different Limbs .................................. 118 7.5 Composition of Homeomorphisms and Tuning ................. 121 8 Repelling Dynamics at Misiurewicz Points 123 8.1 Expanding Homeomorphisms at Misiurewicz Points .............. 123 8.2 α- and β-Type Misiurewicz Points ....................... 126 8.3 Homeomorphisms at Endpoints, Homeomorphisms Between Edges ..... 129 8.4 Scaling Properties of M at a .......................... 132 8.5 Scaling Properties of Frames and of h ..................... 137 8.6 Scaling Properties of M on Multiple Scales .................. 142 9 Combinatorial Surgery and Homeomorphism Groups 145 9.1 The Mapping of External Angles ........................ 145 9.2 H¨older Continuity of H ............................. 149 9.3 Combinatorial Approach to Surgery ...................... 151 9.4 Homeomorphism Groups of M ......................... 156 9.5 Homeomorphism Groups of S1/∼ ........................ 159 Bibliography 163 Index of Symbols and Definitions 169 5 Abstract 2 Consider the iteration of complex quadratic polynomials fc(z) = z +c. The filled-in Julia set Kc contains all z ∈ C with a bounded orbit. The Mandelbrot set M consists of those parameters c ∈ C, such that Kc is connected. Quasi-conformal surgery in the dynamic plane is employed to obtain homeomorphisms h : EM → EeM between subsets of M. We give a general construction of h under the additional assumption that EM = EeM . Then h has a countable family of mutually homeomorphic fundamental domains. Moreover, it extends to a homeomorphism of C, which is quasi-conformal in the exterior of M. The homeomorphisms h : EM → EM considered here fall into two categories: homeomorphisms on edges and homeomorphisms at Misiurewicz points. Edges EM ⊂ M are constructed combinatorially. For a large class of edges EM , there is an associated homeomorphism h : EM → EM . Many edges consist of homeomorphic building blocks which are called frames. Here we employ families of homeomorphisms, since the frames are finer than the fundamental domains of a single homeomorphism. If a homeomorphism h is fixing a Misiurewicz point a, then h or h−1 will be expanding there, thus defining repelling dynamics in the parameter plane. Mappings with this property are constructed for all α-type and β-type Misiurewicz points, and the relation to the well- known asymptotic self-similarity of M at a is discussed. Moreover, a family of similarities on different scales is obtained. Zusammenfassung Die ausgef¨ullte Juliamenge Kc ist ¨uber die Iteration komplexer quadratischer Polynome 2 der Form fc(z) = z + c definiert: z ∈ C geh¨ort zu Kc , wenn der Orbit von z beschr¨anktist. Die Mandelbrotmenge M enth¨alt die Parameter c ∈ C, f¨urdie Kc zusammenh¨angend ist. Mittels quasikonformer Chirurgie in der Dynamik erh¨altman Hom¨oomorphismen h : EM → EeM zwischen Teilmengen von M. Wir geben eine allgemeine Konstruktion unter der zus¨atzlichen Voraussetzung EM = EeM . Dann hat h eine abz¨ahlbare Familie hom¨omorpher Fundamentalbereiche und setzt sich zu einem Hom¨oomorphismus von C fort, der im Außeren¨ von M quasikonform ist. Wir betrachten zwei Typen von Hom¨oomorphismen h : EM → EM : Hom¨oomorphismen auf Edges (Kanten), und Hom¨oomorphismen an Misiurewicz Punkten. Edges EM ⊂ M werden kombinatorisch konstruiert. F¨urviele Edges EM wird ein Hom¨oo- morphismus h : EM → EM erhalten, und sie bestehen aus hom¨oomorphen Bausteinen, den Frames (Rahmen). Der Beweis erfordert eine Familie von Hom¨oomorphismen, da die Frames kleiner sind als die Fundamentalbereiche einer einzelnen Abbildung. Wenn ein Hom¨oomorphismus h : EM → EM einen Misiurewicz Punkt a festl¨aßt,dann ist h oder h−1 dort expandierend, und definiert somit eine repulsive Dynamik im Parameterbe- reich. Derartige Hom¨oomorphismen werden f¨uralle α-Typ und β-Typ Misiurewicz Punkte konstruiert, und wir untersuchen den Zusammenhang zu der bekannten asymptotischen Selbst¨ahnlichkeit von M an a. Außerdem
Load more

Recommended publications
  • On the Mandelbrot Set for I2 = ±1 and Imaginary Higgs Fields
    Journal of Advances in Applied Mathematics, Vol. 6, No. 2, April 2021 https://dx.doi.org/10.22606/jaam.2021.62001 27 On the Mandelbrot Set for i2 = ±1 and Imaginary Higgs Fields Jonathan Blackledge Stokes Professor, Science Foundation Ireland. Distinguished Professor, Centre for Advanced Studies, Warsaw University of Technology, Poland. Visiting Professor, Faculty of Arts, Science and Technology, Wrexham Glyndwr University of Wales, UK. Professor Extraordinaire, Faculty of Natural Sciences, University of Western Cape, South Africa. Honorary Professor, School of Electrical and Electronic Engineering, Technological University Dublin, Ireland. Honorary Professor, School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, South Africa. Email: [email protected] Abstract We consider the consequence√ of breaking with a fundamental result in complex analysis by letting i2 = ±1 where i = −1 is the basic unit of all imaginary numbers. An analysis of the Mandelbrot set for this case shows that a demarcation between a Fractal and a Euclidean object is possible based on i2 = −1 and i2 = +1, respectively. Further, we consider the transient behaviour associated with the two cases to produce a range of non-standard sets in which a Fractal geometric structure is transformed into a Euclidean object. In the case of the Mandelbrot set, the Euclidean object is a square whose properties are investigate. Coupled with the associated Julia sets and other complex plane mappings, this approach provides the potential to generate a wide range of new semi-fractal structures which are visually interesting and may be of artistic merit. In this context, we present a mathematical paradox which explores the idea that i2 = ±1.
    [Show full text]
  • Arxiv:2105.08654V1 [Math.DS] 18 May 2021
    THE DYNAMICS OF COMPLEX BOX MAPPINGS TREVOR CLARK, KOSTIANTYN DRACH, OLEG KOZLOVSKI, AND SEBASTIAN VAN STRIEN Abstract. In holomorphic dynamics, complex box mappings arise as first return maps to well-chosen domains. They are a generalization of polynomial-like mapping, where the domain of the return map can have infinitely many components. They turned out to be extremely useful in tackling diverse problems. The purpose of this paper is: - To illustrate some pathologies that can occur when a complex box mapping is not induced by a globally defined map and when its domain has infinitely many components, and to give conditions to avoid these issues. - To show that once one has a box mapping for a rational map, these conditions can be assumed to hold in a very natural setting. Thus we call such complex box mappings dy- namically natural. Having such box mappings is the first step in tackling many problems in one-dimensional dynamics. - Many results in holomorphic dynamics rely on an interplay between combinatorial and analytic techniques. In this setting some of these tools are - the Enhanced Nest (a nest of puzzle pieces around critical points) from [KSS1]; - the Covering Lemma (which controls the moduli of pullbacks of annuli) from [KL1]; - the QC-Criterion and the Spreading Principle from [KSS1]. The purpose of this paper is to make these tools more accessible so that they can be used as a `black box', so one does not have to redo the proofs in new settings. - To give an intuitive, but also rather detailed, outline of the proof from [KvS, KSS1] of the following results for non-renormalizable dynamically natural complex box mappings: - puzzle pieces shrink to points, - (under some assumptions) topologically conjugate non-renormalizable polynomials and box mappings are quasiconformally conjugate.
    [Show full text]
  • International Journal of Research in Computer Applications and Robotics Issn 2320-7345 Complex Dynamics of Multibrot Sets Fo
    INTERNATIONAL JOURNAL OF RESEARCH IN COMPUTER APPLICATIONS AND ROBOTICS Vol.2 Issue.4, Pg.: 12-22 April 2014 www.ijrcar.com INTERNATIONAL JOURNAL OF RESEARCH IN COMPUTER APPLICATIONS AND ROBOTICS ISSN 2320-7345 COMPLEX DYNAMICS OF MULTIBROT SETS FOR JUNGCK ISHIKAWA ITERATION 1 2 3 Suman Joshi , Dr. Yashwant Singh Chauhan , Dr. Priti Dimri 1 G. B. Pant Engineering College (Pauri Garhwal),[email protected] 2 G. B. Pant Engineering College (Pauri Garhwal), [email protected] 3G. B. Pant Engineering College (Pauri Garhwal), [email protected] Author Correspondence: G. B. Pant Engineering. College, Pauri Garhwal Uttarakhand, 9990423408, [email protected] Abstract The generation of fractals and study of the dynamics of polynomials is one of the emerging and interesting fields of research nowadays. We introduce in this paper the dynamics of modified multibrot function zd - z + c = 0 for d 2 and applied Jungck Ishikawa Iteration to generate new Relative Superior Mandelbrot sets and Relative Superior Julia sets. We have presented here different characteristics of Multibrot function like its trajectories, its complex dynamics and its behaviour towards Julia set are also discussed. In order to solve this function by Jungck –type iterative schemes, we write it in the form of Sz = Tz, where the function T, S are defined as Tz = zd +c and Sz= z. Only mathematical explanations are derived by applying Jungck Ishikawa Iteration for polynomials in the literature but in this paper we have generated relative Mandelbrot sets and Relative Julia sets. Keywords: Complex dynamics, Relative Superior Mandelbrot set, Relative Julia set, Jungck Ishikawa Iteration 1.
    [Show full text]
  • 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]