Fractals Everywhere
by Barnsley, Michael F.Edition: 2nd
Format: Hardcover
Pub. Date: 1993-06-01
Publisher(s): Elsevier Science Ltd
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Summary
This volume is the second edition of the highly successful Fractals Everywhere. The Focus of this text is how fractal geometry can be used to model real objects in the physical world. This edition of Fractals Everywhere is the most up-to-date fractal textbook available today.Fractals Everywhere may be supplemented by Michael F. Barnsley's Desktop Fractal Design System (version 2.0) with IBM for Macintosh software. The Desktop Fractal Design System 2.0 is a tool for designing Iterated Function Systems codes and fractal images, and makes an excellent supplement to a course on fractal geometry* A new chapter on recurrent iterated function systems, including vector recurrent iterated function systems.* Problems and tools emphasizing fractal applciations.* An all-new answer key to problems in the text, with solutions and hints.
Table of Contents
| Foreword | p. XI |
| Acknowledgments | p. XIII |
| Introduction | p. 1 |
| Metric Spaces; Equivalent Spaces; Classification of Subsets; and the Space of Fractals | p. 5 |
| Spaces | p. 5 |
| Metric Spaces | p. 10 |
| Cauchy Sequences, Limit Points, Closed Sets, Perfect Sets, and Complete Metric Spaces | p. 15 |
| Compact Sets, Bounded Sets, Open Sets, Interiors, and Boundaries | p. 19 |
| Connected Sets, Disconnected Sets, and Pathwise-Connected Sets | p. 24 |
| The Metric Space (H (X), h): The Place Where Fractals Live | p. 27 |
| The Completeness of the Space of Fractals | p. 33 |
| Additional Theorems about Metric Spaces | p. 40 |
| Transformations on Metric Spaces; Contraction Mappings; and the Construction of Fractals | p. 42 |
| Transformations on the Real Line | p. 42 |
| Affine Transformations in the Euclidean Plane | p. 49 |
| Mobius Transformations on the Riemann Sphere | p. 58 |
| Analytic Transformations | p. 61 |
| How to Change Coordinates | p. 68 |
| The Contraction Mapping Theorem | p. 74 |
| Contraction Mappings on the Space of Fractals | p. 79 |
| Two Algorithms for Computing Fractals from Iterated Function Systems | p. 84 |
| Condensation Sets | p. 91 |
| How to Make Fractal Models with the Help of the Collage Theorem | p. 94 |
| Blowing in the Wind: The Continous Dependence of Fractals on Parameters | p. 101 |
| Chaotic Dynamics on Fractals | p. 115 |
| The Addresses of Points on Fractals | p. 115 |
| Continuous Transformations from Code Space to Fractals | p. 122 |
| Introduction to Dynamical Systems | p. 130 |
| Dynamics on Fractals: Or How to Compute Orbits by Looking at Pictures | p. 140 |
| Equivalent Dynamical Systems | p. 145 |
| The Shadow of Deterministic Dynamics | p. 149 |
| The Meaningfulness of Inaccurately Computed Orbits is Established by Means of a Shadowing Theorem | p. 158 |
| Chaotic Dynamics on Fractals | p. 164 |
| Fractal Dimension | p. 171 |
| Fractal Dimension | p. 171 |
| The Theoretical Determination of the Fractal Dimension | p. 180 |
| The Experimental Determination of the Fractal Dimension | p. 188 |
| The Hausdorff-Besicovitch Dimension | p. 195 |
| Fractal Interpolation | p. 205 |
| Introduction: Applications for Fractal Functions | p. 205 |
| Fractal Interpolation Functions | p. 208 |
| The Fractal Dimension of Fractal Interpolation Functions | p. 223 |
| Hidden Variable Fractal Interpolation | p. 229 |
| Space-Filling Curves | p. 238 |
| Julia Sets | p. 246 |
| The Escape Time Algorithm for Computing Pictures of IFS Attractors and Julia Sets | p. 246 |
| Iterated Function Systems Whose Attractors Are Julia Sets | p. 266 |
| The Application of Julia Set Theory to Newton's Method | p. 276 |
| A Rich Source for Fractals: Invariant Sets of Continuous Open Mappings | p. 287 |
| Parameter Spaces and Mandelbrot Sets | p. 294 |
| The Idea of a Parameter Space: A Map of Fractals | p. 294 |
| Mandelbrot Sets for Pairs of Transformations | p. 299 |
| The Mandelbrot Set for Julia Sets | p. 309 |
| How to Make Maps of Families of Fractals Using Escape Times | p. 317 |
| Measures on Fractals | p. 330 |
| Introduction to Invariant Measures on Fractals | p. 330 |
| Fields and Sigma-Fields | p. 337 |
| Measures | p. 341 |
| Integration | p. 344 |
| The Compact Metric Space (P (X), d) | p. 349 |
| A Contraction Mapping on (P (X)) | p. 350 |
| Elton's Theorem | p. 364 |
| Application to Computer Graphics | p. 370 |
| Recurrent Iterated Function Systems | p. 379 |
| Fractal Systems | p. 379 |
| Recurrent Iterated Function Systems | p. 383 |
| Collage Theorem for Recurrent Iterated Function Systems | p. 392 |
| Fractal Systems with Vectors of Measures as Their Attractors | p. 403 |
| References | p. 409 |
| References | p. 412 |
| Selected Answers | p. 416 |
| Index | p. 523 |
| Credits for Figures and Color Plates | p. 533 |
| Table of Contents provided by Syndetics. All Rights Reserved. |
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