Fractals Everywhere

by Barnsley, Michael F.

• ISBN13: 9780120790616
• ISBN10: 0120790610
• Edition: 2nd
• Format: Hardcover
• Copyright: 1993-06-01
• Publisher: Elsevier Science Ltd

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.