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Barnsley M. Fractals Everywhere
Barnsley M.  Fractals Everywhere







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: Fractals Everywhere

: Barnsley M.

:

This volume is the second edition of the highly successful Fractals Everywhere. The Focus of this text is how fractal geometry can be used to model real objects in the physical world.

This edition of Fractals Everywhere is the most up-to-date fractal textbook available today.

Fractals Everywhere may be supplemented by Michael F. Barnsley's Desktop Fractal Design System (version 2.0) with IBM for Macintosh software. The Desktop Fractal Design System 2.0 is a tool for designing Iterated Function Systems codes and fractal images, and makes an excellent supplement to a course on fractal geometry A new chapter on recurrent iterated function systems, including vector recurrent iterated function systems.


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$A\setminus B$, definition      11
$A\subset X$, definition      10
$D_H(A)$, definition      202
$d_H(\mu,\nu)$, definition      355
$f^{-1}$, definition      43
$f^{-1}(A)$ (for a set A), definition      66
$f_{on}$, definition      44
$p_1 \& L$, definition      366
$R_{\theta}$, definition      55
$S_0$, definition      24
$x_f$, definition      73
$\blacksquare$, definition      10
$\bullet$, definition      10
$\chi$ definition      350
$\delta_a(B)$, definition      354
$\emptyset$, definition      10
$\epsilon$-net      21
$\epsilon$-net, definition      20
$\forall$, definition      11
$\in$ , definition      6
$\int_{\mathbb{X}}fd\mu$, definition      354
$\mathbb{C}$, definition      8
$\mathbb{R}$, definition      6
$\mathbb{\bar{C}}$, definition      8
$\mathcal{C}$, definition      44 45
$\mathcal{F}$, definition      342
$\mathcal{H}(X)$, definition      30
$\mathcal{H}(X)$, definition, illustration      30
$\mathcal{M}$, definition      315
$\mathcal{M}(A, p)$, definition      200
$\mathcal{M}(A, p, \epsilon)$, definition      200
$\mathcal{N}(a, \epsilon)$, definition      174
$\Rightarrow$, definition      10
$\Sigma$, definition      10 125
$\sigma$-field, definition      344
$\times$, definition      10
$|A|$, definition      58
$|X|$      definition
+ (for set and real numbers), definition      35
Address      3 118125 129 146 151 277 302 303 365
Address, definition      129
Address, illustration      119 120 121 122 123 124 126
Address, periodic      see Periodic address
Affine transformation      2 43 44 4546 50 5253 54 56 57 71 73 74 75 76 78 97 98 102 110 117 121 123 134 141 146 187 214 221 228 231 236 239 240 244 245 256 275 338 366 378
Affine transformation, definition      50
Affine transformation, illustration      46 50 52 53 54 56 58 73 100 101 102 103 105 107 108 233 Plates affine)
Algorithm      see Fractals Specific
Analytic transformation      6268 75 292
Analytic transformation, definition      66
Andes Girl      379
Andes Girl, illustration      Plate 9.8.7
Arctic Wolf, illustration      Plate 9.8.8
Arctic Wolf, illustration, attractive cycle      148 270 274 320
Arctic Wolf, illustration, definition      137
Attractive fixed point      137 142 148 250
Attractive fixed point, definition      136
Attractive fixed point, illustration      138 149
Attractive periodic point, definition      137
Attractor      8384 85 86 87 91 92 94 96 97 98 101 103 105 110 113 118119 120124 125127 129132 133 134 135 142 148 150151 153 154 155 158 160 170 171 173 184 185 186 187 188 202 203 212213 214 216 217 218 220 225 229 235 236 245 248 254 255 265 269 274 275 283 285 301 302 304 305 306 307 311 312 313 316317 321 331 336 338 341 349 352 353 365 366 372 378
Attractor, definition      82
Attractor, illustration      83 95 102 103 105 106 119 124 189 204 233 313
Attractor, strange, definition      82
Basin of attraction      282 284
Basin of attraction, definition      261
Basin of attraction, illustration      282 284
Biological modelling      4 285 290
Black Forest      379380
Black Forest, illustration      Plates 9.8.119.8.14
Bolzano Weierstrass theorem      21 22
Bolzano Weierstrass theorem, illustration      23
Bolzano Weierstrass theorem, proof      21
Borel field      345 346 347 348 350 352353
Borel field, definition      345
Borel measure      4 346 349 350 352 353 354 371 375
Borel measure, illustration      Plate 9.8.3
Borel measure, normalized      355 356
Borel measure, normalized, definition      355
Borel subset      340 345 346 352 367 370 371
Borel subset, definition      340 345
Boundary      25 56 262 265 266 269 283 299 301 305 306 307 311 314 317 321 328 330
Boundary, definition      24
Boundary, illustration      18 25 26 110 123 261
Bounded      22 173 200 201 203 314
Bounded, definition      20
Bounded, totally      21
Boundedness      17
Box Counting Theorem      176177 179 190 192193 226
Box Counting Theorem, illustration      177 179 180
Box Counting Theorem, proof      177
Cantor set      151 156 165 182 265 275
Cantor set, classical      44 75 83 130 134 147 179 182 187 192 302
Cantor set, classical, illustration      45
Cantor set, illustration      150
Caratheodorys extension theorem      4 347 348
Caratheodorys extension theorem, proof      347 (see also Extension theorem)
Cartesian map      25
Cartesian product      27 126127 234
Cartesian product, definition      10
Castle fractal, IFS code for      188
Castle fractal, IFS code for, illustration      187
Cauchy sequence      18 19 21 22 37 41 46 78 79 94 128 173 221 260 291 354
Cauchy sequence, definition      17
Cauchy sequence, illustration      17 29 37 40
Cauchy subsequence, extension of      35 36
Cauchy subsequence, illustration      37
cell      328
Chaos algorithm, illustration      92
Chaos game      2 3
Chaotic dynamical system      3 165 169170 196 277 278 283
Chaotic dynamical system, definition      168
Chaotic, definition      168
Characteristic function, definition      350
Classical Cantor set      see Cantor set classical
Closed      21 22 24 174 291
Closed, definition      19
Closedness      2 17
Closure      133
Closure, definition      19
Cloud Study (video)      113 379
Cloud Study (video), illustration      Plate 3.11.1
Code space      78 85 123 124 125133 134 143 144 146 147148 153 154 161 163 165 168 171 276277 302 312313 344 346 348 352353 365 366 370
Code space, definition      10 125
Collage      377380
Collage theorem      2 9697 98 99101 111 188 221 275 378 380
Collage theorem for measures      365 366
Collage theorem for measures, illustration      367
Collage theorem for measures, proof      365366
Collage theorem, illustration      98 99 100 103 104 108 109
Collage theorem, proof      111
Compact      21 22 23 24 42 80 94 112 127 128 182 186 217 245 258 261 275 291 345 347
Compact metric      147
Compact metric space      see Metric space compact
Compact, definition      20
Compactness      2 17 24 31 160 173 302
Completeness      2 17
Completeness of the space of Fractals theorem      3739
Completeness of the space of Fractals theorem, proof      3839
Completeness, definition      18 (see also Metric space complete)
Completion, of a measure      349
Completion, of a measure, definition      349
Computer programs, deterministic algorithm      8890 213
Computer programs, escape time algorithm      251252 255 256 280 illustration)
Computer programs, hidden variable fractal interpolation function with random iteration algorithm      237238
Computer programs, invariant measure of an IFS with probabilities      372374
Computer programs, invariant measure of an IFS with probabilities, illustration      374
Computer programs, parameter space coloring      314315 317
Computer programs, random iteration algorithm      9192 213
Computer programs, random iteration algorithm with interpolation and vertical scaling factors      215217
Computer programs, random iteration algorithm with interpolation and vertical scaling factors, illustration      217
Computer programs, web diagram      137139
Condensation      220 279
Condensation set      9396
Condensation set, definition      93
Condensation transformation, definition      93
conformal      9
Connected      29 83 259 302 306 309 316 317 321 330
Connected, definition      25
Connected, illustration      28
Connected, multiply      27 29
Connected, multiply, definition      27
Connected, multiply, illustration      28
Connected, pathwise      27 29 240 245 258 261
Connected, pathwise, definition      25
Connected, pathwise, illustration      110
Connected, simply      27 29
Connected, simply, definition      25 27
Connectedness      see also Connected Connection
Connection, illustration      Plate 8.3.1
continuous      4142 62 63 80 81 111 112 127 128 147 219 223 259 272 291 292 353 356 367
Continuous deformation, definition      27
Continuous function      7 234 236 239 240 242 245 349 356 357 370 371
Continuous function, definition      14
Continuous function, illustration      7
Continuous transformation      125 291 302
Continuous transformation, illustration      150
Continuous, uniformly      112113
Continuous, uniformly, definition      41
Contraction mapping      2 5 7579 80 81 82 85 9394 97 111 135 219 220 356
Contraction mapping theorem      76 219
Contraction mapping theorem, proof      76
Contraction mapping, definition      75
Contraction mapping, illustration      77 220
Contractive transformation      76 80 238
Contractive transformation, illustration      77
Contractivity      76 80 81 82 83 84 85 93 94 96 111 113 127 159 160 165 220
Contractivity factor      237 301 303 311
Contractivity factor, definition      75
Convergence      2 18 21 128 173 258 259 272 280 292
Convergence, definition      17
Convergence, illustration      40 47
Convergent sequences      35 41 85 91 279 353 354 356
Convergent sequences, illustration      84 85 243 244
Coordinate changes      6975 77 107/109
Coordinate changes, illustration      71 72
Coordinate space      69
Coordinate space, illustration      71
Coordinate system      69 70 73 179 193 240 297
Coordinate system, illustration      70
Countable      23 84 125 321
Countable base      345
Countable base, definition      345
Countable, definition      124
Countable, illustration      125
Critical point      63 65 67 68 316
Critical point, definition      48 66
Critical point, illustration      49
Critical value      68
Critical value, definition      48 66
Critical value, illustration      49
CYCLE      170171 277 278
Cycle, definition      136 (see also Attractive cycle Repulsive
Cylinder subset      344
Cylinder subset, definition      344
DATA      192 193196 208 212 214 216 218 220 221 224 225226 236
Data, definition      211
Data, generalized      234 235
Data, generalized, definition      234
Data, illustration      197
Deformation, bounded      1314
Deformation, of space      48 5253 179
Degree, of transformation, definition      47
Delta function, definition      354
Dense      167 168 170 171 277
Dense, definition      167
Deterministic algorithm      2 86 87 88 221
Deterministic algorithm, computer program for      8890 213
Deterministic shift dynamical system, illustration      166
diam(A), definition      200
Diameter, of partition, definition      353
Diffeomorphic      149
Differentiable      149 321
Differentiable, definition      148
Disconnected      29 192 321
Disconnected, definition      25
Disconnected, pathwise      27
Disconnected, pathwise, definition      25
Disconnected, totally      29 83 84 119 121 130 143 147 149 151 155 159 165 167 170 171 185 188 202 203 256 302 306 308 322 323 330 365
Disconnected, totally, definition      25 129
Disconnected, totally, illustration      157
Disconnection, illustration      Plate 8.3.1
dS, definition      24
Dynamical system      4 82 134142 143146 151152 248 252 254 256 257 269 270 274 275 280 301 313314 315 316 317 320 321 324 328 330 331
Dynamical system, definition      134
Dynamical system, sensitive to initial conditions, definition      168
Dynamical system, stretch, squeeze, and bend, illustration      135 136
Dynamical system, transitive, definition      168 (see also Chaotic dynamical system Set Shift Slide Stretch squeeze and Transitive
Dynamical systems, family of, illustration      Plates 8.4.18.4.4
Dynamics, on fractals      3
Eltons ergodic theorem      5 370372 378
EQUIVALENCE      2 3 147 148 151 153 165 166 167 203 290 equivalence Metrically
Escape time algorithm      4 248268 269 273 275 280 285 290 292 294 295 296 322 324
Escape time algorithm, computer program      251252 255 256 280
Escape time algorithm, illustration      253 255 257 261 262 266 267268 270 290 293 294 324328 329 331 332 333 Plates 7.3.1
Euclidean geometry      see Geometry classical
Eventually periodic point, definition      137 (see also Periodic eventually)
Extension Lemma      3637
Extension of a Cauchy subsequence      35 36
Extension theorem, Caratheodorys      4
Family resemblances      330
Feigenbaum ratio      320
Fern      92 170 290
Fern, Black Spleenwort      101102 130
Fern, Black Spleenwort, IFS code for      104
Fern, Black Spleenwort, illustration      103
Fern, IFS code for      87 92 103
Fern, illustration      33 40 92 166 351 Plate
Field      343 344 347 348
Field, definition      342
Field, generated by $\mathcal{G}$, definition      343
Field, illustration      344 (see also Borel field Sigma-field)
Fixed point      74 76 78 80 82 103 111 112 131 153 220 272 275 284 291 357
Fixed point, definition      73
Fixed point, table      105 (see also Attractive fixed point Repulsive
FOLDS      4849
Folds, illustration      48 49 50
Fractal dimension      3 17 147 172206 208 210 214 225 226 226227 229 236 250 254 255 266 283 301 306 307 309 312
Fractal dimension of fractal interpolation functions      225 229
Fractal dimension, calculating      175177
Fractal dimension, definition      172 174 182
Fractal dimension, experimental determination of      190199 231
Fractal dimension, experimental determination of illustration      190 192 193 194 195 196 198 199
Fractal dimension, experimental determination of table      191 194
Fractal dimension, idea behind      174
Fractal dimension, illustration      173 178 179 180 185 187 188 189 204 205 206 233
Fractal function, hidden variable      242 244245
Fractal interpolation      3
Fractal interpolation function      34 208 209210 220 221 223 224 226 229 231
Fractal interpolation function, definition      220
Fractal interpolation function, generalized      235236 240
Fractal interpolation function, generalized, definition      235236
Fractal interpolation function, generalized, illustration      241
Fractal interpolation function, hidden variable      237 239 240
Fractal interpolation function, hidden variable, computer program for      237239
Fractal interpolation function, illustration      210 211 215 222 223 224 225 226 227 230231 232 233
Fractal interpolation, hidden variable, illustration      239
Fractal tree      2 95
Fractal, deterministic, definition      80
Fractals, algorithms for computing      2 8696 Deterministic Random
Function      see also Specific functions
Function, characteristic, definition      350
1 2 3
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