Barnsley M. Superfractals

Cambridge University Press, 2006. — 464 p.The chaos game.
Attractors of iterated function systems.
Another chaos game.
Codes, metrics and topologies.Points and spaces.
Functions, mappings and transformations.
Addresses and code spaces.
Metric spaces.
Metrics on code space.
Cauchy sequences, limits and continuity.
Topological spaces.
Important basic topologies.
Some key topological invariants.
Compact sets and spaces.
The Hausdorff metric.
The metric spaces (H(X), dH), (H(H(X)), dH(H)),
Fractal dimensions.
Transformations of points, sets, pictures and measures.Transformations of pictures.
Transformations of measures.
Fixed points and fractals.
Linear and affine transformations in two and three dimensions.
Mobius transformations.
Projective transformations.
Transformations on code spaces.
Semigroups on sets, measures and pictures.Semigroups.
Semigroups of transformations.
Orbits of sets under IFS semigroups.
Orbits of pictures under IFS semigroups.
Orbits of measures under IFS semigroups.
Groups of transformations.
Hyperbolic IFSs, attractors and fractal tops.Hyperbolic IFSs.
The set attractor and the measure attractor.
FS codes.
The chaos game.
FS colouring of set attractors.
The collage theorem.
Deterministic calculation of attractors.
Fractal tops.
Pictures of tops: colour-stealing.
The tops dynamical system.
The fractal top is the fixed point of FTOP.
Relationship between fractal tops and some orbital pictures.
The fractal homeomorphism theorem.
Fractal transformations.
Directed IFSs and general deterministic fractals.
The top of a directed IFS.
A very special case: S: → is open.
Invariant measures for tops dynamical systems.
Superfractals.Computational experiment: glimpse of a superfractal.
SuperIFSs and superfractals.
I-variable IFSs.
The set attractor A(1) of the 1-variable IFS F(1).
Chaos game reveals 1-variable fractal sets.
Hausdorff dimension of some 1-variable fractal sets.
The underlying IFS of a superIFS.
Tops of 1-variable fractal sets.
Homeomorphisms between 1-variable fractal sets and between their tops.
Other sets of 1-variable fractal objects.
V-variable IFSs.
V-variable pictures with stolen colours, and V-variable orbital pictures.
V-variable fractal interpolation.
V-variable space-filling curves.
Fractal transformations between the elements of V-variable superfractals of ‘maybe-not-tops’.
The superfractal of V-variable fractal measures.
Code trees and (general) V-variability.
V-variability and what happens as V →∞.
Final section.