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).
Daugiau apie elektronines knygas