| Foreword |
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xi | |
| Acknowledgments |
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xiii | |
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1 | (4) |
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Metric Spaces; Equivalent Spaces; Classification of Subsets; and the Space of Fractals |
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5 | (37) |
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5 | (5) |
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10 | (5) |
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Cauchy Sequences, Limit Points, Closed Sets, Perfect Sets, and Complete Metric Spaces |
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15 | (4) |
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Compact Sets, Bounded Sets, Open Sets, Interiors, and Boundaries |
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19 | (5) |
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Connected Sets, Disconnected Sets, and Pathwise-Connected Sets |
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24 | (3) |
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The Metric Space (H(X), h): The Place Where Fractals Live |
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27 | (6) |
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The Completeness of the Space of Fractals |
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33 | (7) |
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Additional Theorems about Metric Spaces |
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40 | (2) |
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Transformations on Metric Spaces; Contraction Mappings; and the Construction of Fractals |
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42 | (73) |
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Transformations on the Real Line |
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42 | (7) |
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Affine Transformations in the Euclidean Plane |
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49 | (9) |
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Mobius Transformations on the Riemann Sphere |
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58 | (3) |
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61 | (7) |
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How to Change Coordinates |
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68 | (6) |
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The Contraction Mapping Theorem |
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74 | (5) |
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Contraction Mappings on the Space of Fractals |
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79 | (5) |
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Two Algorithms for Computing Fractals from Iterated Function Systems |
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84 | (7) |
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91 | (3) |
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How to Make Fractal Models with the Help of the Collage Theorem |
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94 | (7) |
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Blowing in the Wind: The Continuous Dependence of Fractals on Parameters |
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101 | (14) |
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Chaotic Dynamics on Fractals |
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115 | (56) |
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The Addresses of Points on Fractals |
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115 | (7) |
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Continuous Transformations from Code Space to Fractals |
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122 | (8) |
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Introduction to Dynamical Systems |
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130 | (10) |
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Dynamics on Fractals: Or How to Compute Orbits by Looking at Pictures |
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140 | (5) |
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Equivalent Dynamical Systems |
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145 | (4) |
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The Shadow of Deterministic Dynamics |
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149 | (9) |
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The Meaningfulness of Inaccurately Computed Orbits is Established by Means of a Shadowing Theorem |
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158 | (6) |
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Chaotic Dynamics on Fractals |
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164 | (7) |
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171 | (34) |
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171 | (9) |
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The Theoretical Determination of the Fractal Dimension |
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180 | (8) |
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The Experimental Determination of the Fractal Dimension |
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188 | (7) |
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The Hausdorff-Besicovitch Dimension |
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195 | (10) |
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205 | (41) |
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Introduction: Applications for Fractal Functions |
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205 | (3) |
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Fractal Interpolation Functions |
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208 | (15) |
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The Fractal Dimension of Fractal Interpolation Functions |
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223 | (6) |
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Hidden Variable Fractal Interpolation |
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229 | (9) |
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238 | (8) |
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246 | (48) |
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The Escape Time Algorithm for Computing Pictures of IFS Attractors and Julia Sets |
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246 | (20) |
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Iterated Function Systems Whose Attractors Are Julia Sets |
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266 | (10) |
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The Application of Julia Set Theory to Newton's Method |
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276 | (11) |
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A Rich Source for Fractals: Invariant Sets of Continuous Open Mappings |
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287 | (7) |
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Parameter Spaces and Mandelbrot Sets |
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294 | (36) |
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The Idea of a Parameter Space: A Map of Fractals |
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294 | (5) |
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Mandelbrot Sets for Pairs of Transformations |
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299 | (10) |
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The Mandelbrot Set for Julia Sets |
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309 | (8) |
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How to Make Maps of Families of Fractals Using Escape Times |
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317 | (13) |
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330 | (49) |
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Introduction to Invariant Measures on Fractals |
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330 | (7) |
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337 | (4) |
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341 | (3) |
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344 | (5) |
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The Compact Metric Space (P(X), d) |
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349 | (1) |
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A Contraction Mapping on (P(X)) |
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350 | (14) |
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364 | (6) |
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Application to Computer Graphics |
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370 | (9) |
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Recurrent Iterated Function Systems |
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379 | (33) |
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379 | (4) |
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Recurrent Iterated Function Systems |
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383 | (9) |
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Collage Theorem for Recurrent Iterated Function Systems |
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392 | (11) |
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Fractal Systems with Vectors of Measures as Their Attractors |
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403 | (6) |
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409 | (3) |
| References |
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412 | (4) |
| Selected Answers |
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416 | (107) |
| Index |
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523 | (10) |
| Credits for Figures and Color Plates |
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533 | |
