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Fractals Everywhere - 2nd Edition - ISBN: 9780120790616, 9781483257693

Fractals Everywhere

2nd Edition

Author: Michael F. Barnsley
eBook ISBN: 9781483257693
Imprint: Academic Press
Published Date: 1st January 1993
Page Count: 548
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Fractals Everywhere, Second Edition covers the fundamental approach to fractal geometry through iterated function systems. This 10-chapter text is based on a course called "Fractal Geometry", which has been taught in the School of Mathematics at the Georgia Institute of Technology. After a brief introduction to the subject, this book goes on dealing with the concepts and principles of spaces, contraction mappings, fractal construction, and the chaotic dynamics on fractals. Other chapters discuss fractal dimension and interpolation, the Julia sets, parameter spaces, and the Mandelbrot sets. The remaining chapters examine the measures on fractals and the practical application of recurrent iterated function systems. This book will prove useful to both undergraduate and graduate students from many disciplines, including mathematics, biology, chemistry, physics, psychology, mechanical, electrical, and aerospace engineering, computer science, and geophysical science.

Table of Contents



Chapter I Introduction

Chapter II Metric Spaces; Equivalent Spaces; Classification of Subsets; And the Space of Fractals

1. Spaces

2. Metric Spaces

3. Cauchy Sequences, Limit Points, Closed Sets, Perfect Sets, and Complete Metric Spaces

4. Compact Sets, Bounded Sets, Open Sets, Interiors, and Boundaries

5. Connected Sets, Disconnected Sets, and Pathwise-Connected Sets

6. The Metric Space (H(X), h):The Place Where Fractals Live

7. The Completeness of the Space of Fractals

8. Additional Theorems about Metric Spaces

Chapter III Transformations on Metric Spaces; Contraction Mappings; And the Construction of Fractals

1. Transformations on the Real Line

2. Affine Transformations in the Euclidean Plane

3. Möbius Transformations on the Riemann Sphere

4. Analytic Transformations

5. How to Change Coordinates

6. The Contraction Mapping Theorem

7. Contraction Mappings on the Space of Fractals

8. Two Algorithms for Computing Fractals from Iterated Function Systems

9. Condensation Sets

10. How to Make Fractal Models with the Help of the Collage Theorem

11. Blowing in the Wind: The Continous Dependence of Fractals on Parameters

Chapter IV Chaotic Dynamics on Fractals

1. The Addresses of Points on Fractals

2. Continuous Transformations from Code Space to Fractals

3. Introduction to Dynamical Systems

4. Dynamics on Fractals: Or How to Compute Orbits by Looking at Pictures

5. Equivalent Dynamical Systems

6. The Shadow of Deterministic Dynamics

7. The Meaningfulness of Inaccurately Computed Orbits is Established by Means of a Shadowing Theorem

8. Chaotic Dynamics on Fractals

Chapter V Fractal Dimension

1. Fractal Dimension

2. The Theoretical Determination of the Fractal Dimension

3. The Experimental Determination of the Fractal Dimension

4. The Hausdorff-Besicovitch Dimension

Chapter VI Fractal Interpolation

1. Introduction: Applications for Fractal Functions

2. Fractal Interpolation Functions

3. The Fractal Dimension of Fractal Interpolation Functions

4. Hidden Variable Fractal Interpolation

5. Space-Filling Curves

Chapter VII Julia Sets

1. The Escape Time Algorithm for Computing Pictures of IFS Attractors and Julia Sets

2. Iterated Function Systems Whose Attractors Are Julia Sets

3. The Application of Julia Set Theory to Newton's Method

4. A Rich Source for Fractals: Invariant Sets of Continuous Open Mappings

Chapter VIII Parameter Spaces and Mandelbrot Sets

1. The Idea of a Parameter Space: A Map of Fractals

2. Mandelbrot Sets for Pairs of Transformations

3. The Mandelbrot Set for Julia Sets

4. How to Make Maps of Families of Fractals Using Escape Times

Chapter IX Measures on Fractals

1. Introduction to Invariant Measures on Fractals

2. Fields and Sigma-Fields

3. Measures

4. Integration

5. The Compact Metric Space (P(X), d)

6. A Contraction Mapping on (P(X))

7. Elton's Theorem

8. Application to Computer Graphics

Chapter X Recurrent Iterated Function Systems

1. Fractal Systems

2. Recurrent Iterated Function Systems

3. Collage Theorem for Recurrent Iterated Function Systems

4. Fractal Systems with Vectors of Measures as Their Attractors

5. References


Selected Answers


Credits for Figures and Color Plates


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© Academic Press 1993
1st January 1993
Academic Press
eBook ISBN:

About the Author

Michael F. Barnsley

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