Fractals Everywhere

ForewordAcknowledgmentsChapter I IntroductionChapter 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 SpacesChapter 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 ParametersChapter 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 FractalsChapter 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 DimensionChapter 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 CurvesChapter 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 MappingsChapter 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 TimesChapter 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 GraphicsChapter 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. ReferencesReferencesSelected AnswersIndexCredits for Figures and Color Plates