Ohrimenko+ / Barnsley. Superfractals

data compression, 188 decreasing sequence, 21 density function, 109

deterministic algorithm, 234, 330

deterministic fractal set, 119, 370, 372, 375, 440 deterministic graph-directed IFS, 370 deterministic iteration algorithm, 324

digital imagery, 423

digital picture, 91, 99, 101, 110 dilated set, 68

dilations, 68

directed 1-variable IFS, 414 directed IFS, 370, 379, 382 directrix, 60

discrete topology, 41, 42 distance between sets, 63 distance function, 23 distinct points, 10

diversity of orbital picture, 264 domain, 14

of picture, 92

duality transformation, 172

dynamical system, 209, 222, 243, 244, 248, 251

embedding, 15, 30 entropy, 244, 298

equivalent metric spaces, 27 equivalent metrics, 27 ergodicity, 42, 203

‘escape time’ algorithm, 222 euclidean distance, 73 euclidean geometry, 298 euclidean metric, 59 euclidean plane, 11 evolution operator, 210 extended complex plane, 142 extended real plane, 140 extension lemma, 78

falling leaves theorem, 85, 86 field, 104

finite binary string, 17 finite covering, 55 finite measure, 107

finitely generated group, 291 fixed point, 115

flower, Morning Glory, 190, 250, 275 flower sets, 82

focus of panabola, 60 fractal

approximation, 71 compression, 237

dimension, 27, 88, 188, 300, 301

invariant under metric transformation, 40 fern, 4, 344, 357

geometry, 100, 207 homeomorphism theorem, 364

homeomorphisms, 4, 358, 370 image compression, 378, 397 interpolation, 426

measure, 128 set, 121 tiling, 218

top, 3, 4, 5, 193, 194, 207, 313, 334, 338, 354, 357, 370, 379, 383

transformation, 326, 366 function, one-to-one, 15

fundamental theorem of affine geometry, 137 fundamental theorem of Mobius¨ transformations, 147 fundamental theorem of projective geometry, 158 furthest-distance function, 70, 71, 73

general V -variable code tree, 436 general V -variable subset, 436, 437 generalized circles, 142 generalized triangles, 305 geometrical properties, 295 geometry, 295

global segments, 234 graph, 13

graph-directed IFS, 379, 382 group, 27, 289

group of transformations, 288, 290, 295 group theory, 293

growth rate for periodic cycles, 244 growth rate of diversity, 224 growth rate of periodic orbits, 298

Hausdorff dimension, 87, 88, 372, 441 Hausdorff distance, 66, 69, 72, 74, 256, 328 Hausdorff measure, 88

Hausdorff metric, 57, 65, 66, 316, 392 Hausdorff space, 39, 41, 45

height of digital image, 99 Heighway dragon, 317

Henon transformation, 222, 275, 276 homeomorphism, 39

hyperbolic fixed point, 165 hyperbolic geometry, 304 hyperbolic IFS, 316

with probabilities, 394

hyperbolic Mobius¨ transformation, 146, 148 hyperbolic tiling, 305

identification topology, 9, 45, 46, 47, 48, 49, 172, 362, 373

identity transformation, 290

IFS, iterated function system, 1, 2, 121, 208, 314 directed, 370, 379, 382

just-touching, 319 linked, 344

local, 370, 378 overlapping, 319, 361 recurrent, 370, 376, 379

totally disconnected, 319 underlying, 398

IFS code affine, 320

hyperbolic, 321 Mobius,¨ 322

Mobius¨ hyperbolic, 322 probabilistic, 321 projective hyperbolic, 321

IFS colour code, 325 IFS group, 291

IFS objects, 293, 295

IFS semigroup, 193, 208, 213 IFS semigroup tiling, 218

IFS with probabilities, 280, 315 image approximation, 327 image compression, 195

image-modelling applications of continuity property, 218

index, 15

infinite strings, 17 information-carrying capacity, 187 information theory, 186, 244, 352 inner product, 133

input screens, 388 interior, 54 intersection, 15

intrinsic Markov chain of order r, 382 invariant measure, 114, 136

invariant picture, 95, 136 invariant probability measure, 324 invariant property, 40

invariant set, 94, 136 inversion, 142 inversive geometry, 304

iterated function system (IFS), 1, 2, 121, 208, 314

Julia set, 196, 209, 222

Klein’s programme, 194, 295, 312

La Hire’s theorem, 181 layered orbit, 219

leaf sets, 82 level set, 58, 63

level-k branch, 435 lexicographic ordering, 20 lifted IFS, 338

limbs of code tree, 435 limit of sequence, 33 limit set, 22 Lindemeyer, 278

line at infinity, 91, 152 linear spaces, 131

linear transformation, 131, 207 linked IFS, 344

Lipschitz constant, 116

Lipschitz transformation, 116, 208 local IFS, 378

local IFS theory, 370

lower left corner of picture, 93

loxodromic Mobius¨ transformation, 146, 148

mapping, 14 Markov chain, 379

Markov Chain Monte Carlo (MCMC) algorithm, 323 Markov operator, 283

Markov process, 186, 203, 375 measurable set, 111

measure, 102, 107 transformation of, 111

measure attractor, 3, 317, 128, 318 meristem, 9, 49, 364

metric, 8, 23 metric space, 23, 24

complete, 33

metric transformation, 27 modular group, 306

Monge–Kantorovitch metric, 125, 323, 328, 331, 392, 441

Morning Glory flower, 190, 250, 275 ‘move-three-points’ algorithm, 139, 160 Mobius¨ geometry, 298, 303, 312 Mobius¨ IFS, 342

Mobius¨ rotation, 115 Mobius¨ strip, 49, 363

Mobius¨ transformation, 89, 92, 97, 112, 140, 142, 145, 148, 149, 151, 167, 181, 195, 207, 292, 296

natural topology, 38, 42, 44, 48, 49, 362, 373 neighbourhood of a point, 38

nested sequence of segments, 256 nodal flip transformations, 189 non-degenerate conic section, 164 non-euclidean geometry, 92 normalized measure, 107

one-parameter semigroup, 210 one-to-one function, 15 1-variable fractal measure, 409 1-variable fractal set, 393 1-variable IFS, 393

1-variable measure IFS, 409

1-variable orbital picture, 411, 412

1-variable picture, 414

1-variable superfractal, 393

1-variable underneath picture, 413 onto function, 15

open covering, 55 open mapping, 39 open set, 38

open transformation, 375 optical algorithm, 69

optimization, 71 orbit, 209

of measure, 280 of picture, 224 of point, 211

of semigroup, 211 orbit stealing, 278 orbital measure, 192, 282

orbital picture, 4, 192, 193, 223, 226, 243, 245, 248, 265, 308, 354, 357

with code space structure, 297 orbital set, 192, 215

orbital tiling, 224 orbit of a set, 214

orthogonal transformation, 133 outer measure, 110

output screens, 388 overlapping IFS, 361

panel, 242, 248, 296 panelling, 270

conjugate, 299 Pappus’ theorem, 181

parabolic Mobius¨ transformation, 146 path of steepest descent, 59, 328 pathwise connected space, 50

pattern matching, 71

perfect topological space, 50 period of periodic point, 186 periodic point, 186, 244 permutation group, 290

on code space, 312 perspective transformation, 153 perspectivity, 150

phase portrait, 99, 210 picture, 177, 247

of orbit of picture, 226 picture function, 92, 93, 188 picture segment, 199 picture tile

address of, 268 generator of, 268

picture tiling, 266, 268, 272

pixel, 101, 109, 167, 176, 177, 324, 334 pixel-chaining algorithm, 378

pixel field, 105

pixel function, 42, 99 pixel σ -algebra, 105, 109 point at infinity, 140 points, distinct, 10 positive measure, 107 preserved property, 40

probability measure, 107, 206 product topology, 43, 44 projective geometry, 92, 298, 301 projective plane, 49, 152, 168, 172

projective transformation, 89, 92, 149, 151, 154, 156, 159, 168, 169, 173, 177, 179, 181, 195, 207, 296

push-forward, 401

quantized colours, 114

random fractal sets, 385, 440, 441 random iteration, 274

random iteration algorithm, 202, 234, 323 random orbit, 394, 400, 412

range of a function, 14

rate of growth of diversity, 264 recurrent IFS, 370, 376, 379

regionally transitive symbolic dynamic system, 383 relative topology, 42

repeller, 277 reptiles, fractal, 270

resolution of digital picture, 99 restricted code structure, 381, 403

homeomorphic, 381, 403 restricted tops code space, 381, 403 restriction, 16

Riemann sphere, 91, 144, 152, 209, 222, 289, 296 Riemannian metric, 146

saturated colour, 114 segment of picture, 199 segments

closure of, 256 space of, 255

self-referential equation, 192, 216, 227, 235, 296, 316

self-similar spaces, 12 semigroup, 197

continuous, 211 discrete, 211

of linear transformations, 207 of Mobius¨ transformations, 207

of projective transformations, 207 of transformations, 206

on code spaces, 208 semigroup tile, 218, 268 semigroup tiling, 196, 266 sequentially compact space, 55 set attractor, 3, 121, 316, 370

set of 1-variable orbital pictures, 412 set of 1-variable tops, 400

shear transformation, 137 shield space, 23

shield subsets, 82 shields, 22

shift-invariant subset of code space, 372 shift-invariant subspace, 186

shift transformation, 185, 187, 188, 241 shortest-distance function, 58, 59, 73 Sierpinski group, 305