SuperFractals

SUPERFRACTALS: PATTERNS OF NATURE......Page 1
Half-title......Page 2
Title Page......Page 4
Copyright Page......Page 5
Dedication......Page 6
Contents......Page 8
Acknowledgements......Page 10
0.1 The chaos game......Page 12
0.2 Attractors of iterated function systems......Page 13
0.3 Another chaos game......Page 16
1.1 Introduction......Page 19
1.2 Points and spaces......Page 21
1.3 Functions, mappings and transformations......Page 24
1.4 Addresses and code spaces......Page 27
1.5 Metric spaces......Page 34
1.6 Metrics on code space......Page 39
1.7 Cauchy sequences, limits and continuity......Page 44
1.8 Topological spaces......Page 48
1.9 Important basic topologies......Page 52
1.10 Some key topological invariants......Page 60
1.11 Compact sets and spaces......Page 65
1.12 The Hausdorff metric......Page 68
1.13 The metric spaces (ℍ(X), d ℍ), (ℍ(ℍ(X)), d ℍ(ℍ)),.........Page 89
1.14 Fractal dimensions......Page 98
2.1 Introduction......Page 100
2.2 Transformations of pictures......Page 103
2.3 Transformations of measures......Page 112
2.4 Fixed points and fractals......Page 126
2.5 Linear and affine transformations in two and three dimensions......Page 140
2.6 Möbius transformations......Page 151
2.7 Projective transformations......Page 160
2.8 Transformations on code spaces......Page 194
3.1 Introduction......Page 201
3.2 Semigroups......Page 208
3.3 Semigroups of transformations......Page 217
3.4 Orbits of sets under IFS semigroups......Page 225
3.5 Orbits of pictures under IFS semigroups......Page 234
3.6 Orbits of measures under IFS semigroups......Page 290
3.7 Groups of transformations......Page 299
4.1 Introduction......Page 324
4.2 Hyperbolic IFSs......Page 325
4.3 The set attractor and the measure attractor......Page 327
4.4 IFS codes......Page 330
4.5 The chaos game......Page 334
4.6 IFS colouring of set attractors......Page 336
4.7 The collage theorem......Page 338
4.8 Deterministic calculation of attractors......Page 341
4.9 Fractal tops......Page 347
4.10 Pictures of tops: colour-stealing......Page 352
4.11 The tops dynamical system......Page 357
4.12 The fractal top is the fixed point of ℱ TOP......Page 363
4.13 Relationship between fractal tops and some orbital pictures......Page 365
4.14 The fractal homeomorphism theorem......Page 368
4.15 Fractal transformations......Page 376
4.16 Directed IFSs and general deterministic fractals......Page 381
4.17 The top of a directed IFS......Page 391
4.18 A very special case: S : Ω → Ω is open......Page 393
4.19 Invariant measures for tops dynamical systems......Page 394
5.1 Introduction......Page 396
5.2 Computational experiment: glimpse of a superfractal......Page 397
5.3 SuperIFSs and superfractals......Page 402
5.4 1-variable IFSs......Page 403
5.5 The set attractor A^(1) of the 1-variable IFS ℱ^(1)......Page 404
5.6 Chaos game reveals 1-variable fractal sets......Page 405
5.7 Hausdorff dimension of some 1-variable fractal sets......Page 407
5.8 The underlying IFS of a superIFS......Page 409
5.9 Tops of 1-variable fractal sets......Page 410
5.10 Homeomorphisms between 1-variable fractal sets and between their tops......Page 414
5.11 Other sets of 1-variable fractal objects......Page 418
5.12 V-variable IFSs......Page 426
5.13 V-variable pictures with stolen colours, and V-variable orbital pictures......Page 432
5.14 V-variable fractal interpolation......Page 437
5.15 V-variable space-filling curves......Page 441
5.16 Fractal transformations between the elements of V-variable superfractals of ‘maybe-not-tops’......Page 442
5.17 The superfractal of V-variable fractal measures......Page 444
5.18 Code trees and (general) V-variability......Page 445
5.19 V-variability and what happens as V → ∞......Page 451
5.20 Final section......Page 453
References......Page 454
Index......Page 460