Fractals Everywhere: New Edition
"Difficult concepts are introduced in a clear fashion with excellent diagrams and graphs." — Alan E. Wessel, Santa Clara University
"The style of writing is technically excellent, informative, and entertaining." — Robert McCarty
This new edition of a highly successful text constitutes one of the most influential books on fractal geometry. An exploration of the tools, methods, and theory of deterministic geometry, the treatment focuses on how fractal geometry can be used to model real objects in the physical world. Two sixteen-page full-color inserts contain fractal images, and a bonus CD of an IFS Generator provides an excellent software tool for designing iterated function systems codes and fractal images.
Suitable for undergraduates and graduate students of many backgrounds, the treatment starts with an introduction to basic topological ideas. Subsequent chapters examine transformations on metric spaces, dynamics on fractals, fractal dimension and interpolation, Julia sets, and parameter spaces. A final chapter introduces measures on fractals and measures in general. Problems and tools emphasize fractal applications, and an answers section contains solutions and hints.
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The Addresses of Points on Fractals
Continuous Transformations from Code Space to Fractals
Introduction to Dynamical Systems
Or How to Compute Orbits
Möbius Transformations on the Riemann Sphere
How to Change Coordinates
The Contraction Mapping Theorem
Contraction Mappings on the Space of Fractals
Two Algorithms for Computing Fractals from Iterated Function Systems
How to Make Fractal Models with the Help of the Collage Theorem
The Continous Dependence of Fractals on Parameters
addresses affine transformations approximately attractor ball Barnsley Borel measure boundary Cantor set Cauchy sequence Chapter code space Collage Theorem compact metric space completes the proof connected continuous function contraction mapping contractivity factor converges coordinates corresponding defined Definition deterministic distance dynamical system associated equation Escape Time Algorithm Euclidean metric Examples & Exercises family of dynamical filled Julia set finite fixed point follows fractal dimension fractal geometry fractal interpolation function fractal system graph Hence homeomorphism hyperbolic illustrated in Figure interval invariant measure invertible iterated function systems just-touching Lemma Let denote limit point Mandelbrot set Markov operator metric equivalence Möbius transformation nonempty number of iterations numits open sets parameter space picture plane Program radius Random Iteration Algorithm real numbers recurrent set of points shift dynamical system Show shown in Figure Sierpinski triangle similitude sphere Suppose symbols Theorem 2.1 totally disconnected