Hausdorff dimension of biaccessible angles of quadratic Julia sets and of the Mandelbrot set Henk Bruin (University of Surrey) joint work with Dierk
|
|
- Vernon Briggs
- 5 years ago
- Views:
Transcription
1 Hausdorff dimension of biaccessible angles of quadratic Julia sets and of the Mandelbrot set Henk Bruin (University of Surrey) joint work with Dierk Schleicher (Jacobsuniversität Bremen) 1
2 2 Personae Dramatis: Quadratic polynomials f c (z) = z 2 + c; Basin of : B c ( ) = {z C : f n (z) }; Julia set J c = B c ( ); Mandelbrot set M = {c C : J c is connected}; Riemann map for dynamic plane ϕ c : B c ( ) {z C : z > 1}; R c (ϑ) = ϕ 1 c ({re 2πiϑ : r > 1}) is dynamic ray of angle ϑ; Riemann map for parameter plane ϕ : C \ M {z C : z > 1}; R(ϑ) = ϕ 1 ({re 2πiϑ : r > 1}) is parameter ray of angle ϑ.
3 Figure 1. Left: the kneading sequence of an external angle ϑ (here ϑ = 1/6) is defined as the itinerary of the orbit of ϑ under angle doubling, where the itinerary is taken with respect to the partition formed by the angles ϑ/2, and (ϑ + 1)/2. Right: in the dynamics of a polynomial for which the ϑ-ray lands at the critical value, an analogous partition is formed by the dynamic rays at angles ϑ/2 and (ϑ + 1)/2, which land together at the critical point.
4 4 Dendritic model for Julia sets For every c M, there is a model (J, f) that is combinatorially representing what happens on the true Julia set. J is a dendrite: compact, one-dimensional connected, locally connected space without loops. f : J J is continuous and unicritical: there is a unique critical point 0 such that f is locally homeomorphic away from 0. Notation for critical orbit: c i = f i (0). 0 separates J into two components J 1 c 1 and J 0, and f : J e J is onto for e {0, 1}. f : J J is transitive (hence no superfluous arms). A point x J has valency n if J \ {x} has n components. biaccessible { 1 end point 2 3 branch point
5 5 Symbolic dynamics: Using the partition J = J 0 J 1 {0} we can define symbolic dynamics. The itinerary of z J is 0 if f i (z) J 0 ; e(z) = e 0 e 1 e 2... where e i = 1 if f i (z) J 1 ; if f i (z) = 0; The kneading sequence ν = ν 1 ν 2 ν 3... is the itinerary of the critical point (neglecting the initial ). The ρ-function is defined as ρ = ρ ν : N N { }, ρ ν (n) = inf{k > n : ν k ν k n }. The internal address is the ρ-orbit of 1. 1 = S 0 S 1 S 2... More generally, for fixed itinerary x and kneading sequence ν, define also ρ ν,x (n) := min{k > n : x k ν k n }.
6 6 Biaccessibility: A point is biaccessible if at least two external rays land on it. An angle ϑ is biaccessible if there is another angle ϑ so that their external rays land at the same point. This definition holds both for dynamic and parameter space. Modulo local connectivity, c M is biaccessible if and only if c J c is biaccessible. Theorem 1. Let ν be the kneading sequence of a Julia set J and z J have itinerary e(z) = x. Then the valency of J at z is equal to the number of grand ρ ν,x -orbits. Modulo local connectivity: the valency of c in M is equal to the number of grand ρ-orbits.
7 c 6 c c 3 c 5 8 p 1 p 4 p 0 p 0 p 6 p 3 c 0 = c 13 p 5 c 1 c w c 10 4 c 7 c 2 p 2 c 9 c 11 c 12 7 Figure 2. The Hubbard tree for with orb(p 0 ) drawn in and pre-periodic branch point w [p 0, c 7 ]; it maps to p 1 in three iterates. Hubbard trees: The Hubbard tree (T, f) of a Julia set (or more easily, its dendritic model) is the connected hull of the critical orbit. f(t ) = T, but f : T T is not globally 2-to-1 (except for z z 2 2). c 1 is always an endpoint of T ; so 0 has at most two branches in T. Every biaccessible point in J eventually maps into T.
8 8 Admissibility: For every ν = 1ν 2 ν 3 {0, 1} N 0, there is a dendritic model J and Hubbard tree T such that its kneading sequence is ν. However, not very such model belongs to a true Julia set of a quadratic polynomial. This is not is much about local connectivity, but rather about the existence of evil branch points, that is: m- periodic branch points so that f m does fix one of its arms, and permutes the others. c 1 c 5 c 4 c 3 c 0 = c 6 c 2 Figure 3. The Hubbard tree for contains an evil orbit of period 3. The Admissibility Condition A kneading sequence ν {0, 1} N 0 fails the admissibility condition (for period m) if the following three conditions hold: (1) the internal address of ν does not contain m; (2) if k < m divides m, then ρ(k) m; (3) ρ(m) < and if r {1,..., m} is congruent to ρ(m) modulo m, then orb ρ (r) contains m. Theorem 2. The non-admissible kneading sequences are dense. The admissible kneading sequences have positive ( 1 2, 1 2 )-Bernoulli measure. Corollary 1. Restricting to admissible kneading sequences has no real effect on Hausdorff dimension of biaccessible parameter angles.
9 9 Hausdorff estimate near the antenna of M Let N := 1 + min{i > 1 : ν i = 1} log(2 N/2 1 1) if N {6, 7, 8,... } log 2 L 1 (N) := N/2 1 1/2 if N = 5, 0 if N {3, 4} U 1 (N) := log(2n 1) log 2 N, and L 1 (N) = U 1 (N) = 1 if N =. Theorem 3. (Julia set) For every external parameter angle ϑ S 1, the set of biaccessible external dynamic angles ϕ S 1 has Hausdorff dimension between L 1 (N) and U 1 (N) for N = N(ϑ). In particular, the set of biaccessible external dynamic angles has Hausdorff dimension less than 1 unless ϑ = 1/2. Theorem 4. (Mandelbrot set) The set of external parameter angles ϑ for with N(ϑ) = N has Hausdorff dimension in between L 1 (N) and U 1 (N). In particular, the measure of biaccessible parameter angles (and this includes the real spine of M) is zero.
10 10 Hausdorff estimate near the main cardioid of M Define κ := sup{k 1 : S j is a multiple of S j 1 for all 1 j k}, L 2 (κ) := 1 S κ+1 { } 7 U 2 (κ) := min 1, 2(S κ + 1), and if κ =, then L 2 (κ) = U 2 (κ) = 0; if S κ < S κ+1 =, then 0 = L 2 (κ) < U 2 (κ). Theorem 5. (Julia set) For every external parameter angle ϑ S 1, the set of biaccessible external dynamic angles ϕ S 1 has Hausdorff dimension between L 2 (κ) and U 2 (κ) for κ = κ(ϑ). In particular, the set of biaccessible external dynamic angles of the Feigenbaum map has Hausdorff dimension zero. Theorem 6. (Mandelbrot set) The set of external parameter angles ϑ for with κ(ϑ) = κ has Hausdorff dimension in between L 2 (κ) and U 2 (κ). In particular, the Hausdorff dimension of angles of infinitely renormalizable maps is zero.
11 Theorem 7. Let B ϑ be the set of combinatorially biaccessible dynamic external angles for parameter angle ϑ, and (T ϑ, f) is the corresponding (abstract) Hubbard tree. Then dim H (B ϑ ) = h top(f Tϑ ). (1) log 2 Furthermore, these numbers depend Hölder continuously on ϑ S 1, with Hölder exponent equal to dim H (B ϑ ). Remarks: Monotonicity was proven in the Ph.D-theses of Chris Penrose (1994) and Tao Li (2007). (Monotonicity of ϑ h top (f Tϑ ) is in the sense that if ϑ and φ are biaccessible parameter angles such that φ (ϑ, ϑ ), i.e., φ belongs to the wake, or shorter arc, of ϑ and its companion angle ϑ with the same kneading sequence, then h top (f Tφ ) h top (f Tϑ ).) With Hölder exponent dim H (B ϑ ) = h top(f Tϑ ) log 2, Hölder continuity fails at the zero-entropy locus. For estimates of ϑ h top (f c(ϑ) Tϑ ) along the the real antenna (and especially the Feigenbaum parameter), see also work by Carminati and Tiozzo. A similar question which we don t solve here is the modulus of continuity of c h top (f c Tϑ ) along the the real antenna. 11
12 12 Steps in the proof: I: dim H (B ϑ ) = h top (f Tϑ )/ log 2 for finite trees: By Variational Principle. II. Argument is inconclusive for infinite trees: The problem is potential non-compactness of infinite Hubbard trees. taking the closure can increase entropy. III. Continuity of ϑ dim H (B ϑ ): Exploit that for large N, up to a set of tiny Hausdorff dimension, biaccessibility of itineraries can be determined by the first N entries of ν. IV. Monotonicity of ϑ h top (f Tϑ ): Itineraries of Hubbard tree of the smaller parameter angle are represented as Cantor set within the Hubbard tree of the parameter larger. V. The biaccessible Mandelbrot set: Monotonicity + Continuity of Hausdorff dimension implies Continuity of Entropy as long as we have majorizing parameter angles. No majorized angles at tips of the Mandelbrot set. VI. Zero entropy tips: No possibility of jumps here. VII. Continuity of ϑ h top (f Tϑ ) at tips: Compare non-zero entropy tips with nearby Thurston-Misiurewicz tips.
13 13 References [B] Arne Beurling, Ensembles exceptionnels, Acta Math. 72 (1940) [BS1] Henk Bruin, Dierk Schleicher, Admissibility of kneading sequences and structure of Hubbard trees for quadratic polynomials, J. London. Math. Soc. 8 (2009), [BS2] Henk Bruin, Dierk Schleicher, Combinatorial and topological biaccessibility of quadratic polynomials, in preparation. [BKS] Henk Bruin, Alexandra Kaffl, Dierk Schleicher, Existence of quadratic Hubbard trees, Fund. Math. 202 (2009), [Ch] Kai Lai Chung, A course in probability theory, Harcourt, Brace & World, Inc., New York (1968). [CT] C. Carminati, G. Tiozzo, The bifurcation locus of numbers of bo9unded type, Preprint [DH1] Adrien Douady, John Hubbard, Études dynamique des polynômes complexes I & II, Publ. Math. Orsay. ( ) (The Orsay notes). [LS] Eike Lau, Dierk Schleicher, Internal addresses in the Mandelbrot set and irreducibility of polynomials, Stony Brook Preprint #19 (1994). [Mat] Pertti Mattila, Geometry of sets and measures in Euclidean spaces, Cambridge University Press, (1995). [Man] Anthony Manning, Logarithmic capacity and renormalizability for landing on the Mandelbrot set, Bull. London Math. Soc. 28 (1996), [Mi1] John Milnor, Dynamics in one complex variable. Introductory lectures, Friedr. Vieweg & Sohn, Braunschweig (1999). [MS] Philipp Meerkamp, Dierk Schleicher Hausdorff dimension and biaccessibility for polynomial Julia sets, to appear in Proc. Amer. Math. Soc. arxiv: v2 [math.ds] [Pen] Chris Penrose, On quotients of shifts associated with dendrite Julia sets of quadratic polynomials, Ph.D. Thesis, University of Coventry, (1994). [Re] Mary Rees, A partial description of the parameter space of rational maps of degree two: Part 1, Acta. Math. 168 (1992), (See also: Realization of matings of polynomials as rational maps of degree two, preprint 1986). [Sch1] Dierk Schleicher, Internal Addresses in the Mandelbrot Set and Galois Groups of Polynomials. Preprint. [Sch2] Dierk Schleicher, On fibers and local connectivity of Mandelbrot and Multibrot sets. In: M. Lapidus, M. van Frankenhuysen (eds): Fractal Geometry and Applications: A Jubilee of Benoit Mandelbrot. Proceedings of Symposia in Pure Mathematics 72, American Mathematical Society (2004), [SZ] Dierk Schleicher, Saeed Zakeri: On biaccessible points in the Julia set of a Cremer quadratic polynomial. Proc. Amer. Math. Soc (1999), [Shi] Mitsuhiro Shishikura, On a theorem of M. Rees for matings of polynomials, in: Tan Lei (ed.), The Mandelbrot set, theme and variations, Cambridge University Press 274 (2000), [Sm] Stas Smirnov, On support of dynamical laminations and biaccessible points in Julia set, Colloq. Math. 87 (2001), [Ta1] Tan Lei, Similarity between the Mandelbrot set and Julia sets, Commun. Math. Phys. 134 (1990), [Ta2] Tan Lei, Matings of quadratic polynomials, Ergod. Th. & Dynam. Sys. 12 (1992), [TL] Tao Li, A monotonicity conjecture for the entropy of Hubbard trees, PhD thesis, Stony Brook (2007). [Za] Saeed Zakeri, Biaccessibility in quadratic Julia sets, Ergod. Th. & Dynam. Sys. 20 (2000), [Zd] Anna Zdunik, On biaccessible points in Julia sets of polynomials, Fund. Math. 163 (2000),
HAUSDORFF DIMENSION OF BIACCESSIBLE ANGLES FOR QUADRATIC POLYNOMIALS. VERSION OF May 8, Introduction
HAUSDORFF DIMENSION OF BIACCESSIBLE ANGLES FOR QUADRATIC POLYNOMIALS. VERSION OF May 8, 2017 HENK BRUIN AND DIERK SCHLEICHER Abstract. A point c in the Mandelbrot set is called biaccessible if two parameter
More informationTopological entropy of quadratic polynomials and sections of the Mandelbrot set
Topological entropy of quadratic polynomials and sections of the Mandelbrot set Giulio Tiozzo Harvard University Warwick, April 2012 Summary 1. Topological entropy Summary 1. Topological entropy 2. External
More informationAn entropic tour of the Mandelbrot set
An entropic tour of the Mandelbrot set Giulio Tiozzo Harvard University June 21, 2013 Summary 1. Topological entropy Summary 1. Topological entropy 2. External rays Summary 1. Topological entropy 2. External
More informationADMISSIBILITY OF KNEADING SEQUENCES AND STRUCTURE OF HUBBARD TREES FOR QUADRATIC POLYNOMIALS
ADMISSIBILITY OF KNEADING SEQUENCES AND STRUCTURE OF HUBBARD TREES FOR QUADRATIC POLYNOMIALS HENK BRUIN AND DIERK SCHLEICHER Abstract. Hubbard trees are invariant trees connecting the points of the critical
More informationEXISTENCE OF QUADRATIC HUBBARD TREES. 1. Introduction
EXISTENCE OF QUADRATIC HUBBARD TREES HENK BRUIN, ALEXANDRA KAFFL, AND DIERK SCHLEICHER Abstract. A (quadratic) Hubbard tree is an invariant tree connecting the critical orbit within the Julia set of a
More informationThe core entropy of polynomials of higher degree
The core entropy of polynomials of higher degree Giulio Tiozzo University of Toronto In memory of Tan Lei Angers, October 23, 2017 First email: March 4, 2012 Hi Mr. Giulio Tiozzo, My name is Tan Lei.
More informationON BIACCESSIBLE POINTS OF THE MANDELBROT SET
ON BIACCESSIBLE POINTS OF THE ANDELBROT SET SAEED ZAKERI Abstract. This paper provides a description for the quadratic polynomials on the boundary of the andelbrot set which are typical in the sense of
More informationEXTERNAL RAYS AND THE REAL SLICE OF THE MANDELBROT SET
EXTERNAL RAYS AND THE REAL SLICE OF THE MANDELBROT SET SAEED ZAKERI Abstract. This paper investigates the set of angles of the parameter rays which land on the real slice [ 2, 1/4] of the Mandelbrot set.
More informationCONTINUITY OF CORE ENTROPY OF QUADRATIC POLYNOMIALS
CONTINUITY OF CORE ENTROPY OF QUADRATIC POLYNOMIALS GIULIO TIOZZO Abstract. The core entropy of polynomials, recently introduced by W. Thurston, is a dynamical invariant which can be defined purely in
More informationCore entropy and biaccessibility of quadratic polynomials
Core entropy and biaccessibility of quadratic polynomials arxiv:1401.4792v1 [math.ds] 20 Jan 2014 Wolf Jung Gesamtschule Aachen-Brand Rombachstrasse 99, 52078 Aachen, Germany. E-mail: jung@mndynamics.com
More informationA dissertation presented. Giulio Tiozzo. The Department of Mathematics
Entropy, dimension and combinatorial moduli for one-dimensional dynamical systems A dissertation presented by Giulio Tiozzo to The Department of Mathematics in partial fulfillment of the requirements for
More informationCONTINUITY OF CORE ENTROPY OF QUADRATIC POLYNOMIALS
CONTINUITY OF CORE ENTROPY OF QUADRATIC POLYNOMIALS GIULIO TIOZZO arxiv:1409.3511v1 [math.ds] 11 Sep 2014 Abstract. The core entropy of polynomials, recently introduced by W. Thurston, is a dynamical invariant
More informationRational Maps with Cluster Cycles and the Mating of Polynomials
Rational Maps with Cluster Cycles and the Mating of Polynomials Thomas Sharland Institute of Mathematical Sciences Stony Brook University 14th September 2012 Dynamical Systems Seminar Tom Sharland (Stony
More informationNon Locally-Connected Julia Sets constructed by iterated tuning
Non Locally-Connected Julia Sets constructed by iterated tuning John Milnor Stony Brook University Revised May 26, 2006 Notations: Every quadratic map f c (z) = z 2 + c has two fixed points, α and β, where
More informationNotes on Tan s theorem on similarity between the Mandelbrot set and the Julia sets
Notes on Tan s theorem on similarity between the Mandelbrot set and the Julia sets Tomoi Kawahira Abstract This note gives a simplified proof of the similarity between the Mandelbrot set and the quadratic
More informationJULIA SETS AND BIFURCATION DIAGRAMS FOR EXPONENTIAL MAPS
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 11, Number 1, July 1984 JULIA SETS AND BIFURCATION DIAGRAMS FOR EXPONENTIAL MAPS BY ROBERT L. DEVANEY ABSTRACT. We describe some of the
More informationWhat is a core and its entropy for a polynomial?
e core-entropy := d dim(bi-acc(f )) What is a core and its entropy for a polynomial? Core(z 2 + c, c R) := K fc R Core(a polynomial f )=? Def.1. If exists, Core(f ) := Hubbard tree H, core-entropy:= h
More informationEntropy, Dimension and Combinatorial Moduli for One-Dimensional Dynamical Systems
Entropy, Dimension and Combinatorial Moduli for One-Dimensional Dynamical Systems The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters.
More informationPeriodic cycles and singular values of entire transcendental functions
Periodic cycles and singular values of entire transcendental functions Anna Miriam Benini and Núria Fagella Universitat de Barcelona Barcelona Graduate School of Mathematics CAFT 2018 Heraklion, 4th of
More informationOn W. Thurston s core-entropy theory
On W. Thurston s core-entropy theory Bill in Jackfest, Feb. 2011 It started out... in 1975 or so. My first job after graduate school was, I was the assistant of Milnor... At one point, I d gotten a programmable
More informationATTRACTING DYNAMICS OF EXPONENTIAL MAPS
Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 28, 2003, 3 34 ATTRACTING DYNAMICS OF EXPONENTIAL MAPS Dierk Schleicher International University Bremen, School of Engineering and Science Postfach
More informationConnectedness loci of complex polynomials: beyond the Mandelbrot set
Connectedness loci of complex polynomials: beyond the Mandelbrot set Sabyasachi Mukherjee Stony Brook University TIFR, June 2016 Contents 1 Background 2 Antiholomorphic Dynamics 3 Main Theorems (joint
More informationPolynomial Julia sets with positive measure
? Polynomial Julia sets with positive measure Xavier Buff & Arnaud Chéritat Université Paul Sabatier (Toulouse III) À la mémoire d Adrien Douady 1 / 16 ? At the end of the 1920 s, after the root works
More informationCOMBINATORIAL RIGIDITY FOR UNICRITICAL POLYNOMIALS
COMBINATORIAL RIGIDITY FOR UNICRITICAL POLYNOMIALS ARTUR AVILA, JEREMY KAHN, MIKHAIL LYUBICH AND WEIXIAO SHEN Abstract. We prove that any unicritical polynomial f c : z z d + c which is at most finitely
More informationSimple Proofs for the Derivative Estimates of the Holomorphic Motion near Two Boundary Points of the Mandelbrot Set
Simple Proofs for the Derivative Estimates of the Holomorphic Motion near Two Boundary Points of the Mandelbrot Set arxiv:808.08335v [math.ds] 5 Aug 08 Yi-Chiuan Chen and Tomoki Kawahira August 8, 08 Abstract
More informationOn the regular leaf space of the cauliflower
On the regular leaf space of the cauliflower Tomoki Kawahira Department of Mathematics Graduate School of Science Kyoto University Email: kawahira@math.kyoto-u.ac.jp June 4, 2003 Abstract We construct
More informationThe Mandelbrot Set and the Farey Tree
The Mandelbrot Set and the Farey Tree Emily Allaway May 206 Contents Introduction 2 2 The Mandelbrot Set 2 2. Definitions of the Mandelbrot Set................ 2 2.2 The First Folk Theorem.....................
More informationarxiv: v2 [math.ds] 24 Feb 2017
ON THURSTON S CORE ENTROPY ALGORITHM YAN GAO Dedicated to the memory of Tan Lei arxiv:1511.06513v2 [math.ds] 24 Feb 2017 Abstract. The core entropy of polynomials, recently introduced by W. Thurston, is
More informationOn the Dynamics of Quasi-Self-Matings of Generalized Starlike Complex Quadratics and the Structure of the Mated Julia Sets
City University of New York (CUNY) CUNY Academic Works Dissertations, Theses, and Capstone Projects Graduate Center 2009 On the Dynamics of Quasi-Self-Matings of Generalized Starlike Complex Quadratics
More informationIn this paper we consider complex analytic rational maps of the form. F λ (z) = z 2 + c + λ z 2
Rabbits, Basilicas, and Other Julia Sets Wrapped in Sierpinski Carpets Paul Blanchard, Robert L. Devaney, Antonio Garijo, Sebastian M. Marotta, Elizabeth D. Russell 1 Introduction In this paper we consider
More informationCounting Kneading Sequences. Vorrapan Chandee & Tian Tian Qiu
CORNE UNIVERSITY MATHEMATICS DEPARTMENT SENIOR THESIS Counting Kneading Sequences A THESIS PRESENTED IN PARTIA FUFIMENT OF CRITERIA FOR HONORS IN MATHEMATICS Vorrapan Chandee & Tian Tian Qiu May 2004 BACHEOR
More informationSingular Perturbations in the McMullen Domain
Singular Perturbations in the McMullen Domain Robert L. Devaney Sebastian M. Marotta Department of Mathematics Boston University January 5, 2008 Abstract In this paper we study the dynamics of the family
More informationEvolution of the McMullen Domain for Singularly Perturbed Rational Maps
Volume 32, 2008 Pages 301 320 http://topology.auburn.edu/tp/ Evolution of the McMullen Domain for Singularly Perturbed Rational Maps by Robert L. Devaney and Sebastian M. Marotta Electronically published
More informationHyperbolic Component Boundaries
Hyperbolic Component Boundaries John Milnor Stony Brook University Gyeongju, August 23, 2014 Revised version. The conjectures on page 16 were problematic, and have been corrected. The Problem Hyperbolic
More informationOn the topological differences between the Mandelbrot set and the tricorn
On the topological differences between the Mandelbrot set and the tricorn Sabyasachi Mukherjee Jacobs University Bremen Poland, July 2014 Basic definitions We consider the iteration of quadratic anti-polynomials
More informationarxiv: v1 [math.ds] 27 Jan 2015
PREPERIODIC PORTRAITS FOR UNICRITICAL POLYNOMIALS JOHN R. DOYLE arxiv:1501.06821v1 [math.ds] 27 Jan 2015 Abstract. Let K be an algebraically closed field of characteristic zero, and for c K and an integer
More informationRESEARCH STATEMENT GIULIO TIOZZO
RESEARCH STATEMENT GIULIO TIOZZO My research focuses on dynamical systems and their relations with complex analysis, Teichmüller theory, ergodic theory, probability and group theory. My two main lines
More informationMODULI SPACE OF CUBIC NEWTON MAPS
MODULI SPACE OF CUBIC NEWTON MAPS PASCALE ROESCH, XIAOGUANG WANG, AND YONGCHENG YIN arxiv:1512.05098v2 [math.ds] 18 May 2016 Abstract. In this article, we study the topology and bifurcations of the moduli
More informationVisualizing the unit ball for the Teichmüller metric
Visualizing the unit ball for the Teichmüller metric Ronen E. Mukamel October 9, 2014 Abstract We describe a method to compute the norm on the cotangent space to the moduli space of Riemann surfaces associated
More informationLimiting Dynamics of quadratic polynomials and Parabolic Blowups
Limiting Dynamics of quadratic polynomials and Parabolic Blowups John Hubbard Joint work with Ismael Bachy MUCH inspired by work of Adam Epstein and helped by discussions with Xavier Buff and Sarah Koch
More informationarxiv:math/ v1 [math.ds] 15 Jun 1996
Parameter Scaling for the Fibonacci Point arxiv:math/9606218v1 [math.ds] 15 Jun 1996 LeRoy Wenstrom Mathematics Department S.U.N.Y. Stony Brook Abstract We prove geometric and scaling results for the real
More informationSHADOWING AND ω-limit SETS OF CIRCULAR JULIA SETS
SHADOWING AND ω-limit SETS OF CIRCULAR JULIA SETS ANDREW D. BARWELL, JONATHAN MEDDAUGH, AND BRIAN E. RAINES Abstract. In this paper we consider quadratic polynomials on the complex plane f c(z) = z 2 +
More informationTowards Topological Classification Of Critically Finite Polynomials
Towards Topological Classification Of Critically Finite Polynomials SenHu Centre de Mathematiques, Ecole Polytechnique 91128 Palaiseau, France, shu@orphee.polytechnique.jr Yunping Jiang' Dept 01 Mathematics,
More informationHARMONIC MEASURE AND POLYNOMIAL JULIA SETS
HARMONIC MEASURE AND POLYNOMIAL JULIA SETS I. BINDER, N. MAKAROV, and S. SMIRNOV Abstract There is a natural conjecture that the universal bounds for the dimension spectrum of harmonic measure are the
More informationReturn times of polynomials as meta-fibonacci numbers
Return times of polynomials as meta-fibonacci numbers arxiv:math/0411180v2 [mathds] 2 Nov 2007 Nathaniel D Emerson February 17, 2009 University of Southern California Los Angeles, California 90089 E-mail:
More informationFrom Cantor to Semi-hyperbolic Parameters along External Rays
From Cantor to Semi-hyperbolic Parameters along External Rays Yi-Chiuan Chen and Tomoki Kawahira March 9, 208 Abstract For the quadratic family f c (z) = z 2 + c with c in the exterior of the Mandelbrot
More informationarxiv: v1 [math.ds] 12 Nov 2014
SELF-SIMILARITY FOR THE TRICORN HIROYUKI INOU arxiv:1411.3081v1 [math.ds] 12 Nov 2014 Abstract. We discuss self-similar property of the tricorn, the connectedness locus of the anti-holomorphic quadratic
More informationTopology of the regular part for infinitely renormalizable quadratic polynomials
Topology of the regular part for infinitely renormalizable quadratic polynomials Carlos Cabrera and Tomoki Kawahira March 16, 2010 Abstract In this paper we describe the well studied process of renormalization
More informationAntipode Preserving Cubic Maps: the Fjord Theorem
Submitted exclusively to the London Mathematical Society doi:10.1112/0000/000000 Antipode Preserving Cubic Maps: the Fjord Theorem A. Bonifant, X. Buff and John Milnor Abstract This note will study a family
More informationRational Maps with Generalized Sierpinski Gasket Julia Sets
Rational Maps with Generalized Sierpinski Gasket Julia Sets Robert L. Devaney Department of Mathematics Boston University Boston, MA 02215, USA Mónica Moreno Rocha The Fields Institute University of Toronto
More informationGeometric Limits of Julia Sets of Maps z^n + exp(2πiθ) as n
Butler University Digital Commons @ Butler University Scholarship and Professional Work - LAS College of Liberal Arts & Sciences 2015 Geometric Limits of Julia Sets of Maps z^n + exp(2πiθ) as n Scott R.
More informationarxiv:math/ v3 [math.ds] 27 Jun 2006
FILLED JULIA SETS WITH EMPTY INTERIOR ARE COMPUTABLE arxiv:math/0410580v3 [math.ds] 27 Jun 2006 I. BINDER, M. BRAVERMAN, M. YAMPOLSKY Abstract. We show that if a polynomial filled Julia set has empty interior,
More informationDigit Frequencies and Bernoulli Convolutions
Digit Frequencies and Bernoulli Convolutions Tom Kempton November 22, 202 arxiv:2.5023v [math.ds] 2 Nov 202 Abstract It is well known that the Bernoulli convolution ν associated to the golden mean has
More informationThe isomorphism problem of Hausdorff measures and Hölder restrictions of functions
The isomorphism problem of Hausdorff measures and Hölder restrictions of functions Theses of the doctoral thesis András Máthé PhD School of Mathematics Pure Mathematics Program School Leader: Prof. Miklós
More informationThe Dynamics of Two and Three Circle Inversion Daniel M. Look Indiana University of Pennsylvania
The Dynamics of Two and Three Circle Inversion Daniel M. Look Indiana University of Pennsylvania AMS Subject Classification: Primary: 37F10 Secondary: 51N05, 54D70 Key Words: Julia Set, Complex Dynamics,
More informationDENSITY OF ORBITS IN LAMINATIONS AND THE SPACE OF CRITICAL PORTRAITS
DENSITY OF ORBITS IN LAMINATIONS AND THE SPACE OF CRITICAL PORTRAITS ALEXANDER BLOKH, CLINTON CURRY, AND LEX OVERSTEEGEN ABSTRACT. Thurston introduced σ d -invariant laminations (where σ d (z) coincides
More informationPERIODIC POINTS ON THE BOUNDARIES OF ROTATION DOMAINS OF SOME RATIONAL FUNCTIONS
Imada, M. Osaka J. Math. 51 (2014), 215 224 PERIODIC POINTS ON THE BOUNDARIES OF ROTATION DOMAINS OF SOME RATIONAL FUNCTIONS MITSUHIKO IMADA (Received March 28, 2011, revised July 24, 2012) Abstract We
More informationarxiv: v1 [math.gr] 20 Jan 2012
A THOMPSON GROUP FOR THE BASILICA JAMES BELK AND BRADLEY FORREST arxiv:1201.4225v1 [math.gr] 20 Jan 2012 Abstract. We describe a Thompson-like group of homeomorphisms of the Basilica Julia set. Each element
More informationarxiv: v2 [math.ds] 22 Jul 2016
The two-dimensional density of Bernoulli Convolutions Christoph Bandt arxiv:1604.00308v2 [math.ds] 22 Jul 2016 Abstract Bernoulli convolutions form a one-parameter family of self-similar measures on the
More informationDynamical invariants and parameter space structures for rational maps
Boston University OpenBU Theses & Dissertations http://open.bu.edu Boston University Theses & Dissertations 2014 Dynamical invariants and parameter space structures for rational maps Cuzzocreo, Daniel
More informationTopology Proceedings. COPYRIGHT c by Topology Proceedings. All rights reserved.
Topology Proceedings Web: http://topology.auburn.edu/tp/ Mail: Topology Proceedings Department of Mathematics & Statistics Auburn University, Alabama 36849, USA E-mail: topolog@auburn.edu ISSN: 0146-4124
More informationDynamics of Tangent. Robert L. Devaney Department of Mathematics Boston University Boston, Mass Linda Keen
Dynamics of Tangent Robert L. Devaney Department of Mathematics Boston University Boston, Mass. 02215 Linda Keen Department of Mathematics Herbert H. Lehman College, CUNY Bronx, N.Y. 10468 Abstract We
More informationELEMENTARY PROOF OF THE CONTINUITY OF THE TOPOLOGICAL ENTROPY AT θ = 1001 IN THE MILNOR THURSTON WORLD
REVISTA DE LA UNIÓN MATEMÁTICA ARGENTINA Vol. 58, No., 207, Pages 63 88 Published online: June 3, 207 ELEMENTARY PROOF OF THE CONTINUITY OF THE TOPOLOGICAL ENTROPY AT θ = 00 IN THE MILNOR THURSTON WORLD
More informationHyperbolicity of Renormalization for C r Unimodal Maps
Hyperbolicity of Renormalization for C r Unimodal Maps A guided tour Edson de Faria Department of Mathematics IME-USP December 13, 2010 (joint work with W. de Melo and A. Pinto) Ann. of Math. 164(2006),
More informationarxiv: v1 [math.ds] 9 Oct 2015
A CLASSIFICATION OF POSTCRITICALLY FINITE NEWTON MAPS RUSSELL LODGE, YAUHEN MIKULICH, AND DIERK SCHLEICHER arxiv:1510.02771v1 [math.ds] 9 Oct 2015 Abstract. The dynamical classification of rational maps
More informationTHE AREA OF THE MANDELBROT SET AND ZAGIER S CONJECTURE
#A57 INTEGERS 8 (08) THE AREA OF THE MANDELBROT SET AND ZAGIER S CONJECTURE Patrick F. Bray Department of Mathematics, Rowan University, Glassboro, New Jersey brayp4@students.rowan.edu Hieu D. Nguyen Department
More informationCOMBINATORIAL MODELS FOR SPACES OF CUBIC POLYNOMIALS
COMBINATORIAL MODELS FOR SPACES OF CUBIC POLYNOMIALS ALEXANDER BLOKH, LEX OVERSTEEGEN, ROSS PTACEK, AND VLADLEN TIMORIN ABSTRACT. Thurston constructed a combinatorial model for the Mandelbrot set. No combinatorial
More informationDynamics on Hubbard trees
FUNDAME NTA MATHEMATICAE 164(2000) Dynamics on Hubbard trees by Lluís Alsedà and Núria Fagella (Barcelona) Abstract. It is well known that the Hubbard tree of a postcritically finite complex polynomial
More informationPERIODS IMPLYING ALMOST ALL PERIODS FOR TREE MAPS. A. M. Blokh. Department of Mathematics, Wesleyan University Middletown, CT , USA
PERIODS IMPLYING ALMOST ALL PERIODS FOR TREE MAPS A. M. Blokh Department of Mathematics, Wesleyan University Middletown, CT 06459-0128, USA August 1991, revised May 1992 Abstract. Let X be a compact tree,
More informationON KÖNIG S ROOT-FINDING ALGORITHMS.
ON KÖNIG S ROOT-FINDING ALGORITHMS. XAVIER BUFF AND CHRISTIAN HENRIKSEN Abstract. In this article, we first recall the definition of a family of rootfinding algorithms known as König s algorithms. We establish
More informationMath 354 Transition graphs and subshifts November 26, 2014
Math 54 Transition graphs and subshifts November 6, 04. Transition graphs Let I be a closed interval in the real line. Suppose F : I I is function. Let I 0, I,..., I N be N closed subintervals in I with
More informationSecond Julia sets of complex dynamical systems in C 2 computer visualization
Second Julia sets of complex dynamical systems in C 2 computer visualization Shigehiro Ushiki Graduate School of Human and Environmental Studies Kyoto University 0. Introduction As most researchers in
More informationLECTURE 2. defined recursively by x i+1 := f λ (x i ) with starting point x 0 = 1/2. If we plot the set of accumulation points of P λ, that is,
LECTURE 2 1. Rational maps Last time, we considered the dynamical system obtained by iterating the map x f λ λx(1 x). We were mainly interested in cases where the orbit of the critical point was periodic.
More informationTOEPLITZ KNEADING SEQUENCES AND ADDING MACHINES
TOEPLITZ KNEADING SEQUENCES AND ADDING MACHINES LORI ALVIN Department of Mathematics and Statistics University of West Florida 11000 University Parkway Pensacola, FL 32514, USA Abstract. In this paper
More informationTOPOLOGICAL ENTROPY FOR DIFFERENTIABLE MAPS OF INTERVALS
Chung, Y-.M. Osaka J. Math. 38 (200), 2 TOPOLOGICAL ENTROPY FOR DIFFERENTIABLE MAPS OF INTERVALS YONG MOO CHUNG (Received February 9, 998) Let Á be a compact interval of the real line. For a continuous
More informationON NEWTON S METHOD FOR ENTIRE FUNCTIONS
J. London Math. Soc. (2) 75 (2007) 659 676 C 2007 London Mathematical Society doi:10.1112/jlms/jdm046 ON NEWTON S METHOD FOR ENTIRE FUNCTIONS JOHANNES RÜCKERT and DIERK SCHLEICHER Abstract The Newton map
More informationCombinatorial equivalence of topological polynomials and group theory
Combinatorial equivalence of topological polynomials and group theory Volodymyr Nekrashevych (joint work with L. Bartholdi) March 11, 2006, Toronto V. Nekrashevych (Texas A&M) Topological Polynomials March
More informationTan Lei and Shishikura s example of obstructed polynomial mating without a levy cycle.
Tan Lei and Shishikura s example of obstructed polynomial mating without a levy cycle. Arnaud Chéritat CNRS, Univ. Toulouse Feb. 2012 A. Chéritat (CNRS, UPS) Obstructed mating Feb. 2012 1 / 15 Origins
More informationSHADOWING AND INTERNAL CHAIN TRANSITIVITY
SHADOWING AND INTERNAL CHAIN TRANSITIVITY JONATHAN MEDDAUGH AND BRIAN E. RAINES Abstract. The main result of this paper is that a map f : X X which has shadowing and for which the space of ω-limits sets
More informationarxiv: v2 [math.ds] 27 Jan 2016
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000 000 S 0002-9939(XX)0000-0 COMPLEMENTARY COMPONENTS TO THE CUBIC PRINCIPAL HYPERBOLIC DOMAIN arxiv:1411.2535v2 [math.ds] 27
More informationA REMARK ON ZAGIER S OBSERVATION OF THE MANDELBROT SET
Shimauchi, H. Osaka J. Math. 52 (205), 737 746 A REMARK ON ZAGIER S OBSERVATION OF THE MANDELBROT SET HIROKAZU SHIMAUCHI (Received April 8, 203, revised March 24, 204) Abstract We investigate the arithmetic
More informationPERTURBATIONS OF FLEXIBLE LATTÈS MAPS
PERTURBATIONS OF FLEXIBLE LATTÈS MAPS XAVIER BUFF AND THOMAS GAUTHIER Abstract. We prove that any Lattès map can be approximated by strictly postcritically finite rational maps which are not Lattès maps.
More informationThe dynamics of maps close to z 2 + c
0.1 0.05 0.05 0.1 0.15 0.2 0.3 0.2 0.1 0.1 The dynamics of maps close to z 2 + c Bruce Peckham University of Minnesota Duluth Physics Seminar October 31, 2013 Β Overview General family to study: z z n
More informationKneading Sequences for Unimodal Expanding Maps of the Interval
Acta Mathematica Sinica, New Series 1998, Oct., Vol.14, No.4, pp. 457 462 Kneading Sequences for Unimodal Expanding Maps of the Interval Zeng Fanping (Institute of Mathematics, Guangxi University, Nanning
More informationFoliation dynamics, shape and classification
Foliation dynamics, shape and classification Steve Hurder University of Illinois at Chicago www.math.uic.edu/ hurder Theorem: [Denjoy, 1932] There exist a C 1 -foliation F of codimension-1 with an exceptional
More informationKLEINIAN GROUPS AND HOLOMORPHIC DYNAMICS
International Journal of Bifurcation and Chaos, Vol. 3, No. 7 (2003) 959 967 c World Scientific Publishing Company KLEINIAN GROUPS AND HOLOMORPHIC DYNAMICS C. CORREIA RAMOS Departamento de Matemática and
More informationCHAOTIC UNIMODAL AND BIMODAL MAPS
CHAOTIC UNIMODAL AND BIMODAL MAPS FRED SHULTZ Abstract. We describe up to conjugacy all unimodal and bimodal maps that are chaotic, by giving necessary and sufficient conditions for unimodal and bimodal
More informationDISCONNECTED JULIA SETS. Paul Blanchard. Boston University Department of Mathematics Boston, Massachusetts
DISCONNECTED JULIA SETS Paul Blanchard Boston University Department of Mathematics Boston, Massachusetts INTRODUCTION The connectivity properties of the Julia set for a polynomial have an intimate relationship
More informationCentralizers of polynomials
Centralizers of polynomials By ROBERTO TAURASO Abstract. - We prove that the elements of an open dense subset of the nonlinear polynomials set have trivial centralizers, i. e. they commute only with their
More informationarxiv: v1 [math.gn] 29 Aug 2016
A FIXED-POINT-FREE MAP OF A TREE-LIKE CONTINUUM INDUCED BY BOUNDED VALENCE MAPS ON TREES RODRIGO HERNÁNDEZ-GUTIÉRREZ AND L. C. HOEHN arxiv:1608.08094v1 [math.gn] 29 Aug 2016 Abstract. Towards attaining
More informationSierpiński curve Julia sets for quadratic rational maps
Sierpiński curve Julia sets for quadratic rational maps Robert L. Devaney Department of Mathematics Boston University 111 Cummington Street Boston, MA 02215, USA Núria Fagella Dept. de Matemàtica Aplicada
More informationQuadratic Julia Sets with Positive Area
Proceedings of the International Congress of Mathematicians Hyderabad, India, 2010 Quadratic Julia Sets with Positive Area Xavier Buff and Arnaud Chéritat Abstract We recently proved the existence of quadratic
More informationSUBCONTINUA OF FIBONACCI-LIKE INVERSE LIMIT SPACES.
SUBCONTINUA OF FIBONACCI-LIKE INVERSE LIMIT SPACES. H. BRUIN Abstract. We study the subcontinua of inverse limit spaces of Fibonacci-like unimodal maps. Under certain combinatorial constraints no other
More informationarxiv: v1 [math.ds] 13 Oct 2017
INVISIBLE TRICORNS IN REAL SLICES OF RATIONAL MAPS arxiv:1710.05071v1 [math.ds] 13 Oct 2017 RUSSELL LODGE AND SABYASACHI MUKHERJEE Abstract. One of the conspicuous features of real slices of bicritical
More informationAsymptotic composants of periodic unimodal inverse limit spaces. Henk Bruin
Asymptotic composants of periodic unimodal inverse limit spaces. Henk Bruin Department of Mathematics, University of Surrey, Guildford, Surrey GU2 7XH, UK H.Bruin@surrey.ac.uk 1 Unimodal maps, e.g. f a
More informationJoshua N. Cooper Department of Mathematics, Courant Institute of Mathematics, New York University, New York, NY 10012, USA.
CONTINUED FRACTIONS WITH PARTIAL QUOTIENTS BOUNDED IN AVERAGE Joshua N. Cooper Department of Mathematics, Courant Institute of Mathematics, New York University, New York, NY 10012, USA cooper@cims.nyu.edu
More informationDavid E. Barrett and Jeffrey Diller University of Michigan Indiana University
A NEW CONSTRUCTION OF RIEMANN SURFACES WITH CORONA David E. Barrett and Jeffrey Diller University of Michigan Indiana University 1. Introduction An open Riemann surface X is said to satisfy the corona
More informationDYNAMICAL DESSINS ARE DENSE
DYNAMICAL DESSINS ARE DENSE CHRISTOPHER J. BISHOP AND KEVIN M. PILGRIM Abstract. We apply a recent result of the first author to prove the following result: any continuum in the plane can be approximated
More informationTHE CLASSIFICATION OF TILING SPACE FLOWS
UNIVERSITATIS IAGELLONICAE ACTA MATHEMATICA, FASCICULUS XLI 2003 THE CLASSIFICATION OF TILING SPACE FLOWS by Alex Clark Abstract. We consider the conjugacy of the natural flows on one-dimensional tiling
More informationCorrelation dimension for self-similar Cantor sets with overlaps
F U N D A M E N T A MATHEMATICAE 155 (1998) Correlation dimension for self-similar Cantor sets with overlaps by Károly S i m o n (Miskolc) and Boris S o l o m y a k (Seattle, Wash.) Abstract. We consider
More information