Fractals Everywhere

ForewordAcknowlegmentsChapter I - IntroductionChapter II - Metric Spces; Equivalent Spaces: Classification of Subsets; and the Space of Fractals1. Spaces2. Metric Spaces3. Cauchy Sequences, Limit Points, Closed Sets, Perfect Sets, and Complete Metric Spaces 4. Compact Sets, Bounded Sets, Open Sets, Interiors, and Boundaries5. Connected Sets, Disconnected Sets, and Pathwise-Connected Sets6. The Metric Space: The Place where Fractals Live7. The Completeness of the Space of Fractals8. Additional Theorems about Metric SpacesChapter III - Transformations on Metric Spaces; Contraction Mappings: and the Construction of Fractals1. Transformations on the Real Line2. Affine Transformations in the Euclidean Plane3. Mobius Transformations on the Riemann Sphere4. Analytic Transformations5. How to Change Coordinates6. The Contraction Mapping Theorem7. Contraction Mappings on the Space of Fractals8. Two Algorithms for Computing Fractals from Iterated Function Systems9. Condensation Sets10. How to Make Fractal Models with the Help of the Collage Theorem11. Blowing in the Wind: The Continuous Dependence of Fractals on ParametersChapter IV - Chaotic Dynamics on Fractals1. The Address of Points on Fractals2. 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 Dimension1. Fractal Dimension2. The Theoretical Determination of the Fractal Dimension 3. The Experimental Determination of the Fractal Dimension 4. The Hausdorff-Besicovitch DimensionChapter VI - Fractal Interpolation1. Introduction: Applications for Fractal Functions2. Fractal Interpolation Functions 3. The Fractal Dimension of Fractal Interpolation Functions 4. Hidden Variable Fractal Interpolation 5. Space-Filling CurvesChapter VII - Julia Sets1. the Escape Time Algorithm for Computing Pictures of IFS Attractors and Julia Set2. 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 Sets1. The Idea of a Parameter Space: A Map of Fractals2. 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 Fractals1. Introduction to Invariant Measures on Fractals2. 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 3648. Application to Computer GraphicsChapter X - Recurrent Iterated Function Systems1. Fractal Systems2. Recurrent Iterated Function Systems 3. Collage Theorem for Recurrent Iterated Function Systems 4. Fractal Systems with Vectors of Measures as Their Attractors 5. ReferencesReferences Selected Answers Index Credits for Figures and Color Plates