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Computability of Julia Sets / Edition 1
     

Computability of Julia Sets / Edition 1

by Mark Braverman, Michael Yampolsky
 

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ISBN-10: 3540685464

ISBN-13: 9783540685463

Pub. Date: 12/12/2008

Publisher: Springer Berlin Heidelberg

Among all computer-generated mathematical images, Julia sets of rational maps occupy one of the most prominent positions. Their beauty and complexity can be fascinating. They also hold a deep mathematical content.

Overview

Among all computer-generated mathematical images, Julia sets of rational maps occupy one of the most prominent positions. Their beauty and complexity can be fascinating. They also hold a deep mathematical content.

Product Details

ISBN-13:
9783540685463
Publisher:
Springer Berlin Heidelberg
Publication date:
12/12/2008
Series:
Algorithms and Computation in Mathematics Series , #23
Edition description:
2009
Pages:
151
Product dimensions:
6.20(w) x 9.30(h) x 0.60(d)

Table of Contents

1 Introduction to Computability 1

1.1 Discrete computability and complexity 1

1.2 Computability and complexity of real numbers and functions 5

1.3 Computability and complexity of subsets of R[superscript k] 11

1.4 Weakly computable sets 14

1.5 Set-valued functions and uniformity 17

2 Dynamics of Rational Mappings 21

2.1 General facts about Riemann surfaces and the hyperbolic metric 21

2.2 Julia sets of rational mappings 27

3 First Examples 37

3.1 A case study: hyperbolic Julia sets 37

3.2 Maps with parabolic orbits 49

3.3 Computing Julia sets with parabolic orbits efficiently 53

3.4 Lack of uniform computability of Julia sets 60

4 Positive Results 65

4.1 Computability of filled Julia sets 65

4.2 Julia sets without rotation domains 69

4.3 Computable Julia sets of Siegel quadratics 70

4.4 Robust computability 75

5 Negative Results 81

5.1 Siegel disks and Cremer points 81

5.2 Non-computable Julia sets 90

5.3 The complexity of Julia sets 103

5.4 Proofs of the main technical lemmas 107

5.5 Number theory and computability 111

5.6 Quadratics with non-computable Julia sets are rare 113

6 Computability vs Topology 119

6.1 How can the boundary of a computable set be non-computable? 119

6.2 Locally connected quadratic Julia sets 120

6.3 Local connectedness versus computability of J[subscript theta] 127

6.4 Non-computable locally connected Julia sets 136

References 146

Index 149

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