Barnsley Fractals Everywhere

2. Auflage 2014
ISBN: 978-1-4832-5769-3
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 548 Seiten, Web PDF

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.

Autoren/Hrsg.

Barnsley, Michael F.

Inhaltsverzeichnis

1;Front Cover;1
2;Fractals Everywhere;4
3;Copyright Page;5
4;Table of Contents;8
5;Dedication;6
6;Foreword to the Second Edition;12
7;Acknowledgments;14
8;Chapter I. Introduction;16
9;Chapter II. Metric Spaces; Equivalent Spaces; Classification of Subsets; and the Space of Fractals;20
9.1;1 Spaces;20
9.2;2 Metric Spaces;25
9.3;3 Cauchy Sequences, Limit Points, Closed Sets, Perfect Sets, and Complete Metric Spaces;30
9.4;4 Compact Sets, Bounded Sets, Open Sets, and Boundaries;34
9.5;5 Connected Sets, Disconnected Sets, and Pathwise-Connected Sets;39
9.6;6 The Metric Space (H(X),h): The Space Where Fractals Live;42
9.7;7 The Completeness of the Space of Fractals;48
9.8;8 Additional Theorems about Metric Spaces;55
10;Chapter III. Transformations on Metric Spaces; Contraction Mappings; and the Construction of Fractals;57
10.1;1 Transformations on the Real Line;57
10.2;2 Affine Transformations in the Euclidean Plane;64
10.3;3 Möbius Transformations on the Riemann Sphere;73
10.4;4 Analytic Transformations;76
10.5;5 How to Change Coordinates;83
10.6;6 The Contraction Mapping Theorem;89
10.7;7 Contraction Mappings on the Space of Fractals;94
10.8;8 Two Algorithms for Computing Fractals from Iterated Function Systems;99
10.9;9 Condensation Sets;106
10.10;10 How to Make Fractal Models with the Help of the Collage Theorem;109
10.11;11 Blowing in the Wind: The Continuous Dependence of Fractals on Parameters;116
11;Chapter IV. Chaotic Dynamicson Fractals;138
11.1;1 The Addresses of Points on Fractals;138
11.2;2 Continuous Transformations from Code Space to Fractals;145
11.3;3 Introduction to Dynamical Systems;153
11.4;4 Dynamics on Fractals: Or How to Compute Orbits by Looking at Pictures;163
11.5;5 Equivalent Dynamical Systems;168
11.6;6 The Shadow of Deterministic Dynamics;172
11.7;7 The Meaningfulness of Inaccurately Computed Orbits Is Established by Means of a Shadowing Theorem;181
11.8;8 Chaotic Dynamics on Fractals;187
12;Chapter V. Fractal Dimension;194
12.1;1 Fractal Dimension;194
12.2;2 The Theoretical Determination of the Fractal Dimension;203
12.3;3 The Experimental Determination of the Fractal Dimension;211
12.4;4 The Hausdorff-Besicovitch Fractal Dimension;218
13;Chapter VI. Fractal Interpolation;228
13.1;1 Introduction: Applications for Fractal Functions;228
13.2;2 Fractal Interpolation Functions;231
13.3;3 The Fractal Dimension of Fractal Interpolation Functions;246
13.4;4 Hidden Variable Fractal Interpolation;252
13.5;5 Space-Filling Curves;261
14;Chapter VII. Julia Sets;269
14.1;1 The Escape Time Algorithm for Computing Pictures of IFS Attractors and Julia Sets;269
14.2;2 Iterated Function Systems Whose Attractors Are Julia Sets;289
14.3;3 The Application of Julia Set Theory to Newton's Method;299
14.4;4 A Rich Source for Fractals: Invariant Sets of Continuous Open Mappings;310
15;Chapter VIII. Parameter Spaces and Mandelbrot Sets;317
15.1;1 The Idea of a Parameter Space: A Map of Fractals;317
15.2;2 Mandelbrot Sets for Pairs of Transformations;322
15.3;3 The Mandelbrot Set for Julia Sets;332
15.4;4 How to Make Maps of Families of Fractals Using Escape Times;340
16;Chapter IX. Measures on Fractals;353
16.1;1 Introduction to Invariant Measures on Fractals;353
16.2;2 Fields and Sigma-Fields;360
16.3;3 Measures;372
16.4;4 Integration;375
16.5;5 The Compact Metric Space (P(X), d);380
16.6;6 A Contraction Mapping on P(X);381
16.7;7 Elton's Theorem;395
16.8;8 Application to Computer Graphics;401
17;Chapter X. Recurrent Iterated Function Systems;410
17.1;1 Fractal Systems;410
17.2;2 Recurrent Iterated Function Systems;414
17.3;3 Collage Theorem for Recurrent IFS;423
17.4;4 Fractal Systems with Vectors of Measures as Their Attractors;434
17.5;5 References;440
18;References;443
19;Selected Answers;447
20;Index;554
21;Credits for Figures and Color Plates;564