Real Laminations and the Topological Dynamics of Complex Polynomials
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Quasiconformal Mappings, Complex Dynamics and Moduli Spaces
Quasiconformal mappings, complex dynamics and moduli spaces Alexey Glutsyuk April 4, 2017 Lecture notes of a course at the HSE, Spring semester, 2017 Contents 1 Almost complex structures, integrability, quasiconformal mappings 2 1.1 Almost complex structures and quasiconformal mappings. Main theorems . 2 1.2 The Beltrami Equation. Dependence of the quasiconformal homeomorphism on parameter . 4 2 Complex dynamics 5 2.1 Normal families. Montel Theorem . 6 2.2 Rational dynamics: basic theory . 6 2.3 Local dynamics at neutral periodic points and periodic components of the Fatou set . 9 2.4 Critical orbits. Upper bound of the number of non-repelling periodic orbits . 12 2.5 Density of repelling periodic points in the Julia set . 15 2.6 Sullivan No Wandering Domain Theorem . 15 2.7 Hyperbolicity. Fatou Conjecture. The Mandelbrot set . 18 2.8 J-stability and structural stability: main theorems and conjecture . 21 2.9 Holomorphic motions . 22 2.10 Quasiconformality: different definitions. Proof of Lemma 2.78 . 24 2.11 Characterization and density of J-stability . 25 2.12 Characterization and density of structural stability . 27 2.13 Proof of Theorem 2.90 . 32 2.14 On structural stability in the class of quadratic polynomials and Quadratic Fatou Conjecture . 32 2.15 Structural stability, invariant line fields and Teichm¨ullerspaces: general case 34 3 Kleinian groups 37 The classical Poincar´e{Koebe Uniformization theorem states that each simply connected Riemann surface is conformally equivalent to either the Riemann sphere, or complex line C, or unit disk D1. The quasiconformal mapping theory created by M.A.Lavrentiev and C. -
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. -
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. -
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. -
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 . -
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. -
ONE HUNDRED YEARS of COMPLEX DYNAMICS the Subject of Complex Dynamics, That Is, the Behaviour of Orbits of Holomorphic Functions
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by University of Liverpool Repository ONE HUNDRED YEARS OF COMPLEX DYNAMICS MARY REES The subject of Complex Dynamics, that is, the behaviour of orbits of holomorphic functions, emerged in the papers produced, independently, by Fatou and Julia, almost 100 years ago. Although the subject of Dynami- cal Systems did not then have a name, the dynamical properties found for holomorphic systems, even in these early researches, were so striking, so unusually comprehensive, and yet so varied, that these systems still attract widespread fascination, 100 years later. The first distinctive feature of iter- ation of a single holomorphic map f is the partition of either the complex plane or the Riemann sphere into two sets which are totally invariant under f: the Julia set | closed, nonempty, perfect, with dynamics which might loosely be called chaotic | and its complement | open, possibly empty, but, if non-empty, then with dynamics which were completely classified by the two pioneering researchers, modulo a few simply stated open questions. Before the subject re-emerged into prominence in the 1980's, the Julia set was alternately called the Fatou set, but Paul Blanchard introduced the idea of calling its complement the Fatou set, and this was immediately universally accepted. Probably the main reason for the remarkable rise in interest in complex dynamics, about thirty-five years ago, was the parallel with the subject of Kleinian groups, and hence with the whole subject of hyperbolic geome- try. A Kleinian group acting on the Riemann sphere is a dynamical system, with the sphere splitting into two disjoint invariant subsets, with the limit set and its complement, the domain of discontinuity, having exactly similar properties to the Julia and Fatou sets. -
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. -
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. -
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 ................................. -
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. -
Early Days in Complex Dynamics: a History of Complex Dynamics in One Variable During 1906–1942, by Daniel S
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 50, Number 3, July 2013, Pages 503–511 S 0273-0979(2013)01408-3 Article electronically published on April 17, 2013 Early days in complex dynamics: a history of complex dynamics in one variable during 1906–1942, by Daniel S. Alexander, Felice Iavernaro, and Alessandro Rosa, History of Mathematics, 38, American Mathematical Society, Providence, RI, London Mathematical Society, London, 2012, xviii+454 pp., ISBN 978-0- 8218-4464-9, US $99.00 Some time ago, on my way to a special lecture at the Fields Institute, I overheard a couple of graduate students whispering in the corridor: - Are you coming to Shishikura’s talk? -Who’sthat? - The guy who proved that the boundary of the Mandelbrot set has Hausdorff dimension two. - Sweet! That was not the topic of the talk (in fact, HD(∂M) = 2 had been proven 15 years earlier), so it was telling that they even knew what the notions were in the statement. To me, this conversation is symbolic of how much the language of complex dynamics has permeated mathematical culture since the early 1980s, when it was preposterous to think of doing serious research on the properties of quadratic polynomials.1 Back then much of the subject’s popularity had to do with the pretty pictures. A few lines of computer code were enough to start exploring an endless variety of fractal shapes, and ever since then, Julia sets have decorated calculus books, art galleries, and cereal boxes. But there was serious research to be done, and complex dynamics is today a vital branch of dynamical systems that has developed deep connections to differential equations, geometric group theory, harmonic analysis, algebraic number theory, and many other subjects.