Fractals everywhere

This volume is the revised second edition of the original book, published in 1988. It includes additional problems and tools emphasizing fractal applications, as well as an all-new answer key to the text exercises. The revision incorporates new chapters on vector recurrent iterated function systems and application of fractals. It also contains an extended chapter on dynamical systems. Input from advisors and potential adopters should make it more suitable for use as a textbook. This book should be of interest to mathematicians and computer scientists, upper-level undergraduate students and graduate students at universities offering courses in chaos and fractals Part 1 Metric spaces, equivalent spaces, classification of subsets, and the space of fractals: spaces, metric spaces cauchy sequences limit points, closed sets, perfect sets and complete metric spaces compact sets, bounded sets open sets, interiors and boundaries connected sets disconnected sets and pathwise connected sets the metric space (H(X),h) - the place where fractals live the completeness of the space of fractals additional theorems about metric spaces. Part 2 Transformations on metric spaces, contraction mappings and the construction of fractals: transformations on the real line affine transformations in the Euclidean plane mobius transformations on the Riemann sphere analytic transformations, how to change coordinates the contraction mapping theorem contraction mappings on the space of fractals, two algorithms for computing fractals from iterated function systems condensation sets how to make fractal models with the help of the collage theorem blowing in the wind - continuous dependence of fractals on parameters. Part 3 Chaotic dynamics on fractals: the addresses of points on fractals continuous transformations from code space to fractals introduction to dynamical systems dynamics on fractals - or how to compute orbits by looking at pictures equivalent dynamical systems the shadow of deterministic dynamics the meaningfulness of inaccurately computed orbits is established by means of a shadowing theorem chaotic dynamics on fractals. Part 4 Fractal dimension: fractal dimension the theoretical determination of the fractal dimension the experimental determination of the fractal dimension Hausdorff-Besicovitch dimension. Part 5 Fractal interpolation: introduction - applications for fractal functions fractal interpolation functions the fractal dimension of fractal interpolation functions hidden variable fractal interpolation space - filling curves. Part 6 Julia sets: the escape time algorithm for computing pictures of IFS attractors and Julia sets iterated function systems whose attractors are Julia sets the application of Julia set theory to Newton's method a rich source of fractals - invariant sets of continuous open mappings. Part 7 Parameter spaces and mandelbrot sets: the idea of a parameter space - a map of fractals Mandelbrot sets for pairs of transformations the Mandelbrot set for Julia sets how to make maps of families of fractals using escape times. (Part contents). 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. This volume is the second edition of the highly successful Fractals Everywhere. The Focus of this text is how fractal geometry can be used to model real objects in the physical world