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    Fractals Everywhere: New Edition

    4.0 1

    by Michael F. Barnsley, Mathematics

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    Table of Contents

    Introduction to the Dover Edition xi

    Foreword to the Second Edition xv

    Acknowledgments xvii

    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 Möbius 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

    4 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

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    "Difficult concepts are introduced in a clear fashion with excellent diagrams and graphs." — Alan E. Wessel, Santa Clara University
    "The style of writing is technically excellent, informative, and entertaining." — Robert McCarty
    This new edition of a highly successful text constitutes one of the most influential books on fractal geometry. An exploration of the tools, methods, and theory of deterministic geometry, the treatment focuses on how fractal geometry can be used to model real objects in the physical world. Two sixteen-page full-color inserts contain fractal images, and a bonus CD of an IFS Generator provides an excellent software tool for designing iterated function systems codes and fractal images.
    Suitable for undergraduates and graduate students of many backgrounds, the treatment starts with an introduction to basic topological ideas. Subsequent chapters examine transformations on metric spaces, dynamics on fractals, fractal dimension and interpolation, Julia sets, and parameter spaces. A final chapter introduces measures on fractals and measures in general. Problems and tools emphasize fractal applications, and an answers section contains solutions and hints.

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