Invariant factors, Julia equivalences and the (abstract) Mandelbrot set:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
2000
|
| Series: | Lecture notes in mathematics
1732 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | X, 205 S. graph. Darast. |
| ISBN: | 3540674349 |
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MARC
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Record in the Search Index
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| adam_text | Contents
1. Introduction 1
1.1 Quadratic iteration and Julia equivalences 1
1.1.1 Some introductory remarks 1
1.1.2 The framework: Topological dynamics and symbolics.. 3
1.1.3 Prerequisites from quadratic iteration 7
1.1.4 From quadratic Julia sets to Julia equivalences 10
1.1.5 Renormalization of quadratic polynomials (I) 13
1.2 The Mandelbrot set 15
1.2.1 Julia equivalences and the Mandelbrot set (I) 15
1.2.2 The system of hyperbolic components 18
2. Abstract Julia sets 25
2.1 Symbolic dynamics for the angle doubling map 25
2.1.1 Generalized binary expansion 25
2.1.2 Points with periodic kneading sequence 26
2.1.3 Symbolic classification of the points in T 30
2.2 Invariant laminations 32
2.2.1 Some basic geometric statements 32
2.2.2 Invariant laminations constructed from a long chord . 36
2.2.3 The structure graph 40
2.2.4 Construction of further invariant laminations 45
2.3 Julia equivalences 59
2.3.1 From invariant laminations to Julia equivalences 59
2.3.2 The set of all Julia equivalences 65
2.3.3 Periodic equivalence classes, simple closed curves, and
the ramification order 70
3. The Abstract Mandelbrot set 73
3.1 The Abstract Mandelbrot set an atlas of Abstract Julia sets 73
3.1.1 Symbolic description of the Abstract Mandelbrot set.. 73
3.1.2 Periodic points and Lavaurs equivalence relation 77
3.1.3 Kneading sequences, visibility and internal addresses .. 79
3.1.4 Further descriptions of the Abstract Mandelbrot set .. 84
3.2 The ordered Abstract Mandelbrot set 91
X Contents
3.2.1 The subtle structure of «a for periodic a 91
3.2.2 Small copies of Abstract Julia sets and of the Ab¬
stract Mandelbrot set 96
3.2.3 A characterization of points with periodic kneading
sequences 101
3.2.4 Sturmian sequences and angle doubling 102
3.3 Renormalization 107
3.3.1 A symbolic concept for renormalization 107
3.3.2 Renormalization, simple renormalization and tuning .. 112
3.3.3 Further small copies of Abstract Julia sets 115
3.4 Correspondence and Translation Principles 122
3.4.1 Parts of the Abstract Mandelbrot set related to special
Julia equivalences 122
3.4.2 Consequences for the (Abstract) Mandelbrot set 132
4. Abstract and concrete theory 141
4.1 Quadratic iteration 141
4.1.1 Julia equivalences and Julia sets 141
4.1.2 Julia equivalences and the Mandelbrot set (II) 146
4.1.3 Renormalization of quadratic polynomials (II) 162
4.1.4 Non realizable Julia equivalences 169
4.2 Miscellaneous 172
4.2.1 Abstract Julia sets as shift invariant factors 172
4.2.2 Conjugate (Abstract) Julia sets 174
4.2.3 The cardinality of some equivalence classes 178
A. Appendix: Invariant and completely invariant factors 181
A.I Simple statements 181
A.2 Shift invariant factors 184
A.3 Further interesting examples 188
References 193
Index 201
Further symbols used 206
|
| any_adam_object | 1 |
| author | Keller, Karsten |
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| illustrated | Illustrated |
| indexdate | 2024-07-09T18:40:09Z |
| institution | BVB |
| isbn | 3540674349 |
| language | English |
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| series2 | Lecture notes in mathematics |
| spelling | Keller, Karsten Verfasser aut Invariant factors, Julia equivalences and the (abstract) Mandelbrot set Karsten Keller Berlin [u.a.] Springer 2000 X, 205 S. graph. Darast. txt rdacontent n rdamedia nc rdacarrier Lecture notes in mathematics 1732 Julia-Menge Symbolische Dynamik Analytic functions Functions of complex variables Symbolic dynamics Topological dynamics Invarianz (DE-588)4120518-2 gnd rswk-swf Julia-Menge (DE-588)4431306-8 gnd rswk-swf Mandelbrot-Menge (DE-588)4595175-5 gnd rswk-swf Julia-Menge (DE-588)4431306-8 s Mandelbrot-Menge (DE-588)4595175-5 s Invarianz (DE-588)4120518-2 s DE-604 Lecture notes in mathematics 1732 (DE-604)BV000676446 1732 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008970492&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Keller, Karsten Invariant factors, Julia equivalences and the (abstract) Mandelbrot set Lecture notes in mathematics Julia-Menge Symbolische Dynamik Analytic functions Functions of complex variables Symbolic dynamics Topological dynamics Invarianz (DE-588)4120518-2 gnd Julia-Menge (DE-588)4431306-8 gnd Mandelbrot-Menge (DE-588)4595175-5 gnd |
| subject_GND | (DE-588)4120518-2 (DE-588)4431306-8 (DE-588)4595175-5 |
| title | Invariant factors, Julia equivalences and the (abstract) Mandelbrot set |
| title_auth | Invariant factors, Julia equivalences and the (abstract) Mandelbrot set |
| title_exact_search | Invariant factors, Julia equivalences and the (abstract) Mandelbrot set |
| title_full | Invariant factors, Julia equivalences and the (abstract) Mandelbrot set Karsten Keller |
| title_fullStr | Invariant factors, Julia equivalences and the (abstract) Mandelbrot set Karsten Keller |
| title_full_unstemmed | Invariant factors, Julia equivalences and the (abstract) Mandelbrot set Karsten Keller |
| title_short | Invariant factors, Julia equivalences and the (abstract) Mandelbrot set |
| title_sort | invariant factors julia equivalences and the abstract mandelbrot set |
| topic | Julia-Menge Symbolische Dynamik Analytic functions Functions of complex variables Symbolic dynamics Topological dynamics Invarianz (DE-588)4120518-2 gnd Julia-Menge (DE-588)4431306-8 gnd Mandelbrot-Menge (DE-588)4595175-5 gnd |
| topic_facet | Julia-Menge Symbolische Dynamik Analytic functions Functions of complex variables Symbolic dynamics Topological dynamics Invarianz Mandelbrot-Menge |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008970492&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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