9780486488707

Fractals Everywhere New Edition

by
  • ISBN13:

    9780486488707

  • ISBN10:

    0486488705

  • Edition: Revised
  • Format: Paperback
  • Copyright: 2012-07-17
  • Publisher: Dover Publications
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Supplemental Materials

What is included with this book?

  • The New copy of this book will include any supplemental materials advertised. Please check the title of the book to determine if it should include any access cards, study guides, lab manuals, CDs, etc.

Summary

Focusing on how fractal geometry can be used to model real objects in the physical world, this up-to-date edition contains a new chapter on recurrent iterated function systems, problems and tools emphasizing fractal applications, and an answer key. A bonus CD featuring Barnsley's Desktop Fractal Design Systemserves as an excellent supplement. "Technically excellent, informative, and entertaining." Robert McCarty. 32-page full-color insert.

Author Biography

Michael F. Barnsley is a British mathematician, researcher, and author who holds several patents on fractal compression. He is a faculty member of the Mathematical Sciences Institute at Australian National University, and he previously taught in the United States at Georgia Tech.

Table of Contents

Introduction to the Dover Editionp. xi
Foreword to the Second Editionp. xv
Acknowledgmentsp. xvii
Introductionp. 1
Metric Spaces; Equivalent Spaces; Classification of Subsets; and the Space of Fractalsp. 5
Spacesp. 5
Metric Spacesp. 10
Cauchy Sequences, Limit Points, Closed Sets, Perfect Sets, and Complete Metric Spacesp. 15
Compact Sets, Bounded Sets, Open Sets, Interiors, and Boundariesp. 19
Connected Sets, Disconnected Sets, and Pathwise-Connected Setsp. 24
The Metric Space (H(X), h):The Place Where Fractals Livep. 27
The Completeness of the Space of Fractalsp. 33
Additional Theorems about Metric Spacesp. 40
Transformations on Metric Spaces; Contraction Mappings; and the Construction of Fractalsp. 42
Transformations on the Real Linep. 42
Affine Transformations in the Euclidean Planep. 49
Möbius Transformations on the Riemann Spherep. 58
Analytic Transformationsp. 61
How to Change Coordinatesp. 68
The Contraction Mapping Theoremp. 74
Contraction Mappings on the Space of Fractalsp. 79
Two Algorithms for Computing Fractals from Iterated Function Systemsp. 84
Condensation Setsp. 91
How to Make Fractal Models with the Help of the Collage Theoremp. 94
Blowing in the Wind: The Continous Dependence of Fractals on Parametersp. 101
Chaotic Dynamics on Fractalsp. 115
The Addresses of Points on Fractalsp. 115
Continuous Transformations from Code Space to Fractalsp. 122
Introduction to Dynamical Systemsp. 130
Dynamics on Fractals: Or How to Compute Orbits by Looking at Picturesp. 140
Equivalent Dynamical Systemsp. 145
The Shadow of Deterministic Dynamicsp. 149
The Meaningfulness of Inaccurately Computed Orbits is Established by Means of a Shadowing Theoremp. 158
Chaotic Dynamics on Fractalsp. 164
Fractal Dimensionp. 171
Fractal Dimensionp. 171
The Theoretical Determination of the Fractal Dimensionp. 180
The Experimental Determination of the Fractal Dimensionp. 188
The Hausdorff-Besicovitch Dimensionp. 195
Fractal Interpolationp. 205
Introduction: Applications for Fractal Functionsp. 205
Fractal Interpolation Functionsp. 208
The Fractal Dimension of Fractal Interpolation Functionsp. 223
Hidden Variable Fractal Interpolationp. 229
Space-Filling Curvesp. 238
Julia Setsp. 246
The Escape Time Algorithm for Computing Pictures of IFS Attractors and Julia Setsp. 246
Iterated Function Systems Whose Attractors Are Julia Setsp. 266
The Application of Julia Set Theory to Newton's Methodp. 276
A Rich Source for Fractals: Invariant Sets of Continuous Open Mappingsp. 287
Parameter Spaces and Mandelbrot Setsp. 294
The Idea of a Parameter Space: A Map of Fractalsp. 294
Mandelbrot Sets for Pairs of Transformationsp. 299
The Mandelbrot Set for Julia Setsp. 309
How to Make Maps of Families of Fractals Using Escape Timesp. 317
Measures on Fractalsp. 330
Introduction to Invariant Measures on Fractalsp. 330
Fields and Sigma-Fieldsp. 337
Measuresp. 341
Integrationp. 344
The Compact Metric Space (P(X), d)p. 349
A Contraction Mapping on (P(X))p. 350
Elton's Theoremp. 364
Application to Computer Graphicsp. 370
Recurrent Iterated Function Systemsp. 379
Fractal Systemsp. 379
Recurrent Iterated Function Systemsp. 383
Collage Theorem for Recurrent Iterated Function Systemsp. 392
Fractal Systems with Vectors of Measures as Their Attractorsp. 403
Referencesp. 409
Referencesp. 412
Selected Answersp. 416
Indexp. 523
Credits for Figures and Color Platesp. 533
Table of Contents provided by Ingram. All Rights Reserved.

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