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| Barnsley M. — Fractals Everywhere |
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| Предметный указатель |
Function, simple, definition 350 (see also “Simple function”)
Generalized data, definition 234 (see also “Data generalized”)
Geodesies, definition 11
Geometric series 47
Geometric series, illustration 47
Geometry, classical 1 207 208 209 211
Geometry, classical, illustration 116 209
Geometry, deterministic 5 43 239 376
Graph 236 237 240 245 254 275 350
Graph, definition 234
Group 74 75
Group, definition 74
Hausdorff distance 46 80 98 211 292 301
Hausdorff distance, definition 34
Hausdorff distance, illustration 33
Hausdorff metric 35 80 81 88 97 113 115 209 221 272 292 330 365
Hausdorff metric, illustration 114
Hausdorff p-dimensional measure 201
Hausdorff p-dimensional measure, definition 200
Hausdorff p-dimensional measure, illustration 201
Hausdorff — Besicovitch 3
Hausdorff — Besicovitch dimension 200 202 203 204 205 299
Hausdorff — Besicovitch dimension, definition 202
Hausdorff — Besicovitch dimension, illustration 205
Hidden variable 237
Hidden variable, definition 237 (see also “Fractal interpolation function hidden
Homeomorphic 14 146 147 302
Homeomorphic, definition 146
Homeomorphism 16 42 146 147 148 153 167 302
Homeomorphism, definition 14
Homeomorphism, illustration 29 147
Hutchinson metric 355 356
Hutchinson metric, definition 355
IFS 2 83 84 85 86 87 88 96—110 111—117 118—134 142 143—146 147 149 150—151 166 170 184 186 187 202 203 210 213 214—215’ 217—218 220 225 229 235 237 239 244—245 248 255 269—270 273 283 285 311 359
IFS code 221 378 379 380
IFS code for  240 242 245
IFS code for a measure on 335 336 339
IFS code for a measure on , illustration 336 337 338
IFS code for Black Spleen wort 104
IFS code for Castle fractal 188
IFS code for Fern 87 92 103
IFS code for Sierpinski triangle 86 91
IFS code for Square 87
IFS code for Tree, Fractal 87
IFS with condensation, illustration 95 96
IFS with probabilities 336 338 340 348 352 356 364 365 366 370
IFS with probabilities, illustration 368 369
IFS with probabilities, invariant measure of 357
IFS with probabilities, invariant measure of computer program 372—374
IFS with probabilities, invariant measure of computer program, illustration 374 (see also “Iterated function system”)
IFS, definition 82
IFS, hyperbolic 151—159 160—162 165 167 169 171 184—185 187 188 202 204 212 213—214 217—218 221 223 235 245 254 256 275 277 301 302 303 310 316—317 321 356 357 366—367
IFS, hyperbolic with condensation 94 95 122—123
IFS, hyperbolic with condensation, definition 93
IFS, illustration 99 100 101 113 116 189 233
IFS, lifted 156 169
IFS, lifted, definition 155
IFS, random iteration algorithm applied to, computer program 215—216 237—238
Image animation 113
Inf S, definition 35 75
Infinum, definition 35 75
integral 354
Integral of a continuous real-valued function 4
Integral, definition 350 354
Interior 25
Interior, definition 24
Interior, illustration 25 26
Interpolation function 212 213 234 236
Interpolation function, definition 212 234
Interpolation points 219 226 234
Interpolation points, definition 212 234
Interval [a, b], definition 10
Invariant 16 167 283 292
Invariant measure 366—367 370 371 374 378
Invariant measure of IFS with probabilities 365 366 372
Invariant measure of IFS with probabilities, definition 357
Invariant measure of IFS with probabilities, illustration 369
Invariant measure, definition 339—340
Invariant measure, illustration 375 376
Invariant, illustration 15
Invertible 14 49 59 69 151 154—156 159 186 219 302
Invertible transformation 66 74 160
Invertible, definition 43
Iterated function system 2 334
Iterated function system with probabilities 334—335 (see also “IFS”)
Iterated function system with probabilities, definition 334 (see also “IFS with
Iterated function system, definition 82
Iterates 48
Iterates, backward, definition 44
Iterates, forward, definition 44
Jet exhaust 3 196 207—208 209
Jet flame 193—196
Jet flame, illustration 195 196
Julia set 4 62 258 260 265 269 270 272 273 274 275 276 277 278 279 283 284 285 290 301 314 315 316 317 321
Julia set, definition 258 277
Julia set, filled 258 261 262 265 269 273 274 294
Julia set, filled, definition 258
Julia set, filled, illustration 261 262 263 264 314 318 319
Julia set, hyperbolic 273
Julia set, illustration 271 278 279 281 286—289 318 Plates 7.1.9 8.3.1
Just-touching 120 121 129 130 146 150 151—153 159 161 166 185 186 187 202 203 213 227 240 265 277 322 323
Just-touching collage 105—107 115—117
Just-touching collage, illustration 106
Just-touching, definition 129
Just-touching, illustration 152
landscape 379
Landscape, illustration Plates 9.8.1 9.8.2
Leaf 53 98 102—103 105 156 379
Leaf and Sunflower, condensation sets, illustration Plate 9.8.9
Leaf, illustration 54 98 105 340 Plates 9.8.4
Lebesgue measure, definition 349
Lily 328
Lily, illustration 329
lim inf 180
lim sup 180 182 183
LIMIT 18 19 23
Limit point 20
Limit point, definition 19
Limit point, illustration 19
Limit, definition 18
Linear fractional transformation see “Mobius transformation”
Linear space 7 8
Linear transformation 52—53 55 58 79
Linear transformation, illustration 52 53 79
Mandelbrot set 4 303 304 305 306 307 310 311 312 315 316 317 321 328 332
Mandelbrot set, definition 303 315
Mandelbrot set, illustration 304 305 306 307 308 309 310 311 312 316 318 319 320 321 322 323 333 Plate
Mandelbrot, B. 1
MAP 4 82 85 99 102 132 133 144 153 155 161 185 188 202 203 210 212—213 214 236 240 244 245 265 272 273 274 277 283 291 292 297—302 304 307 313 314 320 328 330 366 367 378
Map, illustration 57 58 60 61 67 83 298 299 Plates 8.4.3 8.4.4
Mapping 44 59 60 62 63 65 147 321
Markov operator 356 357 359 364 365 366
Markov operator, definition 356
Markov operator, illustration 358 359 360 361 362 363 364
Measure 4 336 338 341 346 347 348 350 352 356 357
Measure, definition 346
Mercator’s projection 25 280
Mercator’s projection, illustration 282
Metric 12 15 31 79 109 217 220 234 235 275 349 355
Metric equivalence 24 25
Metric space 2 3 11—42 44 69 78 80 81 83 94 113 149 167 168 173 174 175 176 180 182 183 200 201 202 215 219 221 240 291 301 346 347 348 349 350 354
Metric space, closedness of 2 (see also “Closed” “Closedness”)
Metric space, compact 5 41 76 78 79 85 112 113 126 147 275 302 345 349 353 355 356 357 365 370
Metric space, compact, illustration 77
Metric space, compactness of 2 78 “Compactness”)
Metric space, complete 19 21 29 31 33 35 76 82 111 126 127 173 182 218 234 239
Metric space, completeness of 2 (see also “Completeness”)
Metric space, connectedness of 2 (see also “Connected” “Connectedness”)
Metric space, convergence of 2 (see also “Convergence”)
Metric space, definition 11
| Metric space, equivalence of 2 17
Metric space, equivalence of, illustration 22 29
Metric space, equivalent 13 14 15 19 20 21 23 130 132 146 148
Metric space, illustration Plate 2.6.1
Metric space, openness of 2 (see also “Open” “Openness”)
Metric space, subsets of 2 16
Metric space, transformation on 2 (see also “Transformation”)
Metric, compact 147
Metric, definition 11
Metric, equivalent 13 217—218 275
Metric, equivalent, definition 12
Metric, Euclidean 11 13 14 46 58 78 175 176—177 182 200 203—204 213 217—218 221 250 258
Metric, Euclidean, illustration 18
Metric, illustration 12 13 14 18
Metric, Manhattan 11 13 58 78 162 182 211
Metrically equivalent 146 147 148 166 172 179 180 203 302
Metrically equivalent, definition 146
Metrically equivalent, illustration 15
Minimal period, definition 136
Mobius transformation 2 49 50 59—62 73 75 85 134—135 290
Mobius transformation, definition 49 59
Mobius transformation, illustration 60 61 62 138
Model 29 159 173 190 290 314 352 377 378
Monterey Coast 379
Monterey Coast, illustration Plate 9.8.6
Mountain, illustration 107
Myreberg point 320
Natural scene 377
Nested 44 279
Newton transformation 283 284 285 290 320
Newton transformation, definition 230
Newton transformation, illustration 290 Plate
Newton’s method 4 280—290
Non-wavering domain theorem 283
Norm of a point, definition 58
Norm, of the linear transformation, definition 58
Numits 251 254 256 322 323 335 372
Numits, definition 251
One-parameter family 134 141 151 258 274
One-parameter family, illustration 232 233 240
One-to-one 14 44 69 143 147 159
One-to-one, definition 43
Onto 14 44 69 124 127 128
Onto, definition 43
open 22 24 259 291 345
Open set 62 272 292 294
Open set condition, definition 129
Open, definition 21
Openness 2 17
Orbit 3 134 135 137 144 146 147 148 152 154 155 158 159—165 167 168—170 250 257 259 260 272 280 290 292 314 316
Orbit, chaotic 165—166 170
Orbit, definition 134
Orbit, illustration 143 144 145 146 166 167 251
Outer measure, definition 347
overlapping 121 123 129 130 151 152 154 156 163 185 188
Overlapping, definition 121 129
Overlapping, illustration 124 153 157
Parameter space 4 300 301 302 303 310 311 313 314 317 320 321 323 330 331
Parameter space coloring, computer program for 314—315 317
Parameter space, illustration 300 304 Plates
Parameters 74 112 113 214 236 274 303 311 317
Parameters, continuous dependence of fractals on 2 111—117
Parameters, continuous dependence of invariant measure on, illustration Plate 9.8.5
Partition, definition 353
Partition, diameter of, definition 353
Path 25 27
Path, definition 25
Path, illustration 27
Perfect 20 45 84
Perfection 17
Perfection, definition 19
Period, definition 132
Periodic address, definition 132
Periodic point 133 136 277
Periodic point, definition 132 136 “Repulsive
Periodic, eventually 137 142 144 146 148 153
Periodic, eventually, definition 132
Piecewise linear interpolation function 212 214 224
Piecewise linear interpolation function, illustration 212
Point at infinity 257—258 259 260 261 272 275 314 316
Point x, illustration 7 8 18
Polynomial transformation 48 50 68 261 272
Polynomial transformation, definition 47
Polynomial transformation, illustration 47 48 49 50 51
Quadratic transformation 62 63 65—66 68 75 257 259 269 283
Quadratic transformation, illustration 64 65 67
Quadree, definition 122
Quadree, illustration 122
R, definition 55
Random iteration algorithm 86 87 91 134 167 169 170 215 269 273 274 275 335 336 338 339 340 341 352 359 370 371—372 378
Random iteration algorithm, computer program 91—92 213
Random iteration algorithm, computer program, with interpolation and vertical scaling factors 215—216
Random iteration algorithm, illustration 115 270 271 336 337 338 339 340 341
Random shift dynamical system 151 154 158 159 160 161 163 165 169
Random shift dynamical system, definition 154—155
Random shift dynamical system, illustration 154 164
Rational numbers 168
Rational numbers, definition 125
Rational transformation 67 273
Reflection 59
Reflection, definition 55
Repelling sets 4
Repulsive cycle 3 146 148 277
Repulsive cycle, definition 137
Repulsive fixed point 137 144 148 250
Repulsive fixed point, definition 136—137
Repulsive fixed point, illustration 138 149
Repulsive periodic point 137 282
Riemann sphere 8 43 258 272 275 280
Riemann sphere, illustration 9 259 273 276 Plate
Riemann sphere, Mobius transformations on 59—62 64—65
Rotation, definition 55
Scaling factor 184—185 186 188 202 240 265 307 321
Scaling factor, definition 55
Scaling factor, illustration 56 (see also “Vertical scaling factor”)
Semigroup 74 75 78
Semigroup, definition 74
Sensitive to initial conditions, definition 168
Sequence of intervals 46
Sequence of intervals, illustration 46 47
Set dynamical systems, definition 142
Set-valued inverse, definition 66
Seurat 377—380
Shadow theorem 3 156 158—159
Shadow theorem 3, illustration 157 158
Shadowing theorem 3 160 161—162 163 165 170
Shadowing theorem, illustration 163 164
Shadowing theorem, proof 161
Shear transformation 214 215 216—217
Shear transformation, definition 214
Shear transformation, illustration 215 228
Shift dynamical system 3 144 146 147—148 149 151—159 159—161 163 165—167 168 169—170 171 248 250 254—255 256 276 277 278 321—322 331
Shift dynamical system, definition 144
Shift dynamical system, illustration 143 152 153 154 249
Shift dynamical system, lifted, definition 156 (see also “Deterministic shift dynamical system” “Random
Shift operator, definition 74
Shift transformation, definition 144
Sierpinski space , definition 10
Sierpinski triangle 10 41 50 56 63 67 78 85 89 90 92 93 119—120 122 123—124 125 151 153 161 177 186 201 203 245 248—251 252 254 265 292 338 341 346 350—352
Sierpinski triangle, IFS code for 86 91
Sierpinski triangle, illustration 18 51 64 67 89 90 120 150 178 201 247 249 251 339 351
Sierpinski triangle, table 342
Sigma-field 344 345 346 347 348 350
Sigma-field, generated by  , definition 345
Similitude 55 56 59 60 62 63 65—66 68 75 98 99 105 117 184—185 186—188 202 203 240 265 307 321 330
Similitude, definition 54—55
Similitude, illustration 55 64 99 100 105 188
Simple function 350 352 353 354 356
Simple function, definition 350
Simple function, illustration 351
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