Résumé
Table of Contents
- Foreword XI Acknowledgments XIII
Chapter I
- Introduction 1
Chapter II Metric Spaces; Equivalent Spaces; Classification of Subsets; and the Space of Fractals 5
- 1. Spaces 5
- 2. Metric Spaces 10
- 3. Cauchy Sequences, Limit Points, Closed Sets, Perfect Sets, and Complete Metric Spaces 15
- 4. Compact Sets, Bounded Sets, Open Sets, Interiors, and Boundaries 19
- 5. Connected Sets, Disconnected Sets, and Pathwise-Connected Sets 24
- 6. The Metric Space (H (X), h): The Place Where Fractals Live 27
- 7. The Completeness of the Space of Fractals 33
- 8. Additional Theorems about Metric Spaces 40
Chapter III Transformations on Metric Spaces; Contraction Mappings; and the Construction of Fractals 42
- 1. Transformations on the Real Line 42
- 2. Affine Transformations in the Euclidean Plane 49
- 3. Mobius Transformations on the Riemann Sphere 58
- 4. Analytic Transformations 61
- 5. How to Change Coordinates 68
- 6. The Contraction Mapping Theorem 74
- 7. Contraction Mappings on the Space of Fractals 79
- 8. Two Algorithms for Computing Fractals from Iterated Function Systems 84
- 9. Condensation Sets 91
- 10. How to Make Fractal Models with the Help of the Collage Theorem 94
- 11. Blowing in the Wind: The Continous Dependence of
Fractals on Parameters 101
Chapter IV Chaotic Dynamics on Fractals 115
- 1. The Addresses of Points on Fractals 115
- 2. Continuous Transformations from Code Space to Fractals 122
- 3. Introduction to Dynamical Systems 130
- 4. Dynamics on Fractals: Or How to Compute Orbits by Looking at Pictures 140
- 5. Equivalent Dynamical Systems 145
- 6. The Shadow of Deterministic Dynamics 149
- 7. The Meaningfulness of Inaccurately Computed Orbits is Established by Means of a Shadowing Theorem 158
- 8. Chaotic Dynamics on Fractals 164
Chapter V Fractal Dimension 171
- 1. Fractal Dimension 171
- 2. The Theoretical Determination of the Fractal Dimension 180
- 3. The Experimental Determination of the Fractal Dimension 188
- 4. The Hausdorff-Besicovitch Dimension 195
Chapter VI Fractal Interpolation 205
- 1. Introduction: Applications for Fractal Functions 205
- 2. Fractal Interpolation Functions 208
- 3. The Fractal Dimension of Fractal Interpolation Functions 223
- 4. Hidden Variable Fractal Interpolation 229
- 5. Space-Filling Curves 238
Chapter VII Julia Sets 246
- 1. The Escape Time Algorithm for Computing Pictures of IFS Attractors and Julia Sets 246
- 2. Iterated Function Systems Whose Attractors Are Julia Sets 266
- 3. The Application of Julia Set Theory to Newton's Method 276
- 4. A Rich Source for Fractals: Invariant Sets of
Continuous Open Mappings 287
Chapter VIII Parameter Spaces and Mandelbrot Sets 294
- 1. The Idea of a Parameter Space: A Map of Fractals 294
- 2. Mandelbrot Sets for Pairs of Transformations 299
- 3. The Mandelbrot Set for Julia Sets 309 v4. How to
Make Maps of Families of Fractals Using Escape Times 317
Chapter IX Measures on Fractals 330
- 1. Introduction to Invariant Measures on Fractals 330
- 2. Fields and Sigma-Fields 337
- 3. Measures 341
- 4. Integration 344
- 5. The Compact Metric Space (P (X), d) 349
- 6. A Contraction Mapping on (P (X)) 350
- 7. Elton's Theorem 364
- 8. Application to Computer Graphics 370
Chapter X Recurrent Iterated Function Systems 379
- 1. Fractal Systems 379
- 2. Recurrent Iterated Function Systems 383
- 3. Collage Theorem for Recurrent Iterated Function Systems 392
- 4. Fractal Systems with Vectors of Measures as Their Attractors 403
- 5.
References 409
- References 412
- Selected Answers 416
- Index 523
- Credits for Figures and Color Plates 533
Caractéristiques techniques
| PAPIER | |
| Éditeur(s) | Morgan Kaufmann |
| Auteur(s) | Michael F. Barnsley |
| Parution | 01/04/2000 |
| Édition | 2eme édition |
| Nb. de pages | 532 |
| Couverture | Broché |
| Intérieur | Noir et Blanc |
| EAN13 | 9780120790692 |
| ISBN13 | 978-0-12-079069-2 |
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