Homeomorphisms on Edges of the Mandelbrot Set

Von der Fakultät für Mathematik, Informatik und Naturwissenschaften der

Rheinisch-Westfälischen Technischen Hochschule Aachen genehmigte Dissertation

zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften

vorgelegt von

Diplom-Mathematiker Wolf Jung

aus Gelsenkirchen-Buer

Berichter: Universitätsprofessor Dr. Volker Enss

Universitätsprofessor Dr. Gerhard Jank

Universitätsprofessor Dr. Walter Bergweiler

Tag der mündlichen Prüfung: 3.7.2002

Diese Dissertation ist auf den Internetseiten der Hochschulbibliothek online verfügbar.

The author’s present address is:

Wolf Jung

Institut für Reine und Angewandte Mathematik

RWTH Aachen, D-52062 Aachen, Germany

jung@iram.rwth-aachen.de 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

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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, Núria Fagella, Lukas Geyer, Peter Häıssinsky, Heinz Hanßmann,

Karsten Keller, Hartje Kriete, Steffen Rohde, Rudolf Winkel and Tan Lei.

I wish to thank all colleagues at the Institut für Reine und Angewandte Mathematik

for the supporting and warm atmosphere. Part of this work was done on long

evenings in some nice Cafés of Aachen: Kittel, Labyrinth, Last Exit, Meisenfrei,

Molkerei, Orient Expresso, Pontgarten, Wild Roses and Ohne Worte (Munich).

3

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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 Hölder 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

Consider the iteration of complex quadratic polynomials fc(z) = z2 +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 → ẼM between subsets of M. We give a general construction of h under the additional assumption that EM = ẼM . 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 cl