Computability of Julia Sets / Edition 1

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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.
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Editorial Reviews

From the Publisher
From the reviews:
“The study of dynamical systems has at its core … a very computational feel. … One can feel the book trying to be self-contained … . The subject of the book is timely and important. … The questions posed and answered in the present book are natural and the approach well-suited to produce enlightening results. … The book is also generous … . It has the potential to inspire considerable future work in this intriguing field.” (Wesley Calvert, SIGACT News, Vol. 41 (1), 2010)
“Written in an accessible way with many explications, examples and illustrations. … this book sees the meeting of two worlds: computability theory and iteration of rational maps. It is a fruitful one … and a share of surprises. It is also a compendium of several years of research by the authors … together with a lot of new results. … a nice and quick introduction to both topics, and much of it is pleasant to read … . includes interesting discussions and presents stimulating conjectures.” (Arnaud Chéritat, Foundations of Computational Mathematics, Vol. 12, 2012)
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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
  • Edition number: 1
  • Pages: 151
  • Product dimensions: 6.20 (w) x 9.30 (h) x 0.60 (d)

Meet the Author

M. Braverman is an expert in Theoretical Computer Science, particularly in applications of computability to Complex Analysis and Dynamical Systems M. Yampolsky is an expert in Dynamical Systems, particularly in Holomorphic Dynamics and Renormalization Theory

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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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