Computability of Julia Sets / Edition 1

Hardcover (Print)
Buy New
Buy New from BN.com
$79.58
Used and New from Other Sellers
Used and New from Other Sellers
from $54.27
Usually ships in 1-2 business days
(Save 45%)
Other sellers (Hardcover)
  • All (8) from $54.27   
  • New (6) from $54.26   
  • Used (2) from $80.37   

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.
Read More Show Less

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)
Read More Show Less

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

Read More Show Less

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

Read More Show Less

Customer Reviews

Be the first to write a review
( 0 )
Rating Distribution

5 Star

(0)

4 Star

(0)

3 Star

(0)

2 Star

(0)

1 Star

(0)

Your Rating:

Your Name: Create a Pen Name or

Barnes & Noble.com Review Rules

Our reader reviews allow you to share your comments on titles you liked, or didn't, with others. By submitting an online review, you are representing to Barnes & Noble.com that all information contained in your review is original and accurate in all respects, and that the submission of such content by you and the posting of such content by Barnes & Noble.com does not and will not violate the rights of any third party. Please follow the rules below to help ensure that your review can be posted.

Reviews by Our Customers Under the Age of 13

We highly value and respect everyone's opinion concerning the titles we offer. However, we cannot allow persons under the age of 13 to have accounts at BN.com or to post customer reviews. Please see our Terms of Use for more details.

What to exclude from your review:

Please do not write about reviews, commentary, or information posted on the product page. If you see any errors in the information on the product page, please send us an email.

Reviews should not contain any of the following:

  • - HTML tags, profanity, obscenities, vulgarities, or comments that defame anyone
  • - Time-sensitive information such as tour dates, signings, lectures, etc.
  • - Single-word reviews. Other people will read your review to discover why you liked or didn't like the title. Be descriptive.
  • - Comments focusing on the author or that may ruin the ending for others
  • - Phone numbers, addresses, URLs
  • - Pricing and availability information or alternative ordering information
  • - Advertisements or commercial solicitation

Reminder:

  • - By submitting a review, you grant to Barnes & Noble.com and its sublicensees the royalty-free, perpetual, irrevocable right and license to use the review in accordance with the Barnes & Noble.com Terms of Use.
  • - Barnes & Noble.com reserves the right not to post any review -- particularly those that do not follow the terms and conditions of these Rules. Barnes & Noble.com also reserves the right to remove any review at any time without notice.
  • - See Terms of Use for other conditions and disclaimers.
Search for Products You'd Like to Recommend

Recommend other products that relate to your review. Just search for them below and share!

Create a Pen Name

Your Pen Name is your unique identity on BN.com. It will appear on the reviews you write and other website activities. Your Pen Name cannot be edited, changed or deleted once submitted.

 
Your Pen Name can be any combination of alphanumeric characters (plus - and _), and must be at least two characters long.

Continue Anonymously

    If you find inappropriate content, please report it to Barnes & Noble
    Why is this product inappropriate?
    Comments (optional)