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| Ïîèñê ïî óêàçàòåëÿì |
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| Falconer K. — Fractal geometry: mathematical foundations and applications |
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| Ïðåäìåòíûé óêàçàòåëü |
Fractally homogeneous turbulence 309
Fractional Brownian motion 173 267 269 267—271
Fractional Brownian surface(s) 273—275
Fractions, continued 153—154
Frostman's lemma 70
Full square 254
Function(s) 6 6—7
Function(s), continuous 10
Function(s), convex 181 181—182
Functional analysis 179
Gauge function 37
Gaussian distribution 23
Gaussian process 267
General construction 61—62 62
Generator 134—135
Generator, examples 132 133 134
Geometric invariance 41
Geometric measure theory 53 76
Graphs of functions 160 160—169 258 266 267
Gravitational potential 70 306—307
Group of transformations 8 110 111
Group(s) of fractional dimension 182-184
Growth 300—306
Hamilton's equations 207
Hamiltonian 207—208
Hamiltonian systems, stability of 207—208 212
Hausdorff dimension xxiv 27 31 31—33 31 54
Hausdorff dimension and box(-counting) dimension 46 60
Hausdorff dimension and packing dimension 53
Hausdorff dimension and projections 90—93
Hausdorff dimension of a measure 209 288
Hausdorff dimension of attractor 192 193
Hausdorff dimension of self-affine set 140—144 140 144
Hausdorff dimension, Brownian motion 263 265
Hausdorff dimension, calculating 70—72 92
Hausdorff dimension, equivalent definitions 35—36
Hausdorff dimension, fractional Brownian motion 268
Hausdorff dimension, fractional Brownian surface 273
Hausdorff dimension, Levy stable process 272
Hausdorff dimension, properties 32
Hausdorff distance 124 124
Hausdorff measure 14 27—30 28
Hausdorff measure and intersections 113
Hausdorff measure and packing measure 53
Hausdorff measure and product rule 99
Hausdorff measure and quasi-circle 236
Hausdorff metric 124 145 300
Hausdorff — Besicovitch dimension 31
Heat equation 303
Hele-Shaw cell 304
Henon attractor/map 103 196—197 197 198 212 213
Heuristic calculation(s) 34—35 129
Histogram method, for multifractal spectrum 279 283
Hoelder condition 30 32 262 265 268
HoElder exponent 283 312
Hoelder function 8 10 30 161
Hoelder's inequality 297
Homeomorphism 10 33
Homogeneous turbulence 308
Horseshoe map 194—195 195 196 212
Image 7 258 275
Image encoding 145—148
In general 109 110
Independence of events 20
Independence of random variables 20
Independent increment 259 268 311
Index- fractional Brownian function 273
Index- fractional Brownian motion 267 267—271 269 311—312
Index- fractional Brownian surface 273 274
Infimum 5
Information dimension 41
Injection 7
integral 16—17
Integral geometry 118
Interior 6
Interior of loop 223
Intermittency 308
Interpolation 169 170
Intersection formulae 110—113
Intersection(s) 4 109—118 110 265—266 275
Intersection(s), large 113—118 157
interval 4
Interval density 152
Invariance, geometric 41
Invariance, Lipschitz 41 48
Invariant curves 205 207
Invariant measure 208 208—211
Invariant set 123 123—129 187 218
Invariant tori 208
Inverse function 7
Inverse image 7
Investment calculations 186
Irregular point 78
Irregular set 78 79—80 94
Irregular set, examples xxi 81 180
Isolated point, as attractor 201
Isometry 7 30
Isotropic 261
Isotropic turbulence 307
Iterated construction(s) 95—96 96 180—181
Iterated function system (IFS) 123 123—128
Iterated function system (IFS) and repellers 187—189
Iterated function system (IFS), advantages 128
Iterated function system (IFS), attractor for 123 123—129 146—148 194 228
Iterated function system (IFS), variations 135—139
Iterated venetian blind construction 95—96 96 180
Iteration 186—201 215—242
Jarnik's theorem 155—157 205 207
Jordan curve 53 81
Julia set xxii 215 215—242 219 233 234
Kakeya problem 176—179
Koch curve see von Koch curve
Kolmogorov entropy 41
Kolmogorov model of turbulence 307—309
Kolmogorov — Arnold — Moser (KAM) theorem 208
Laminar flow 307
Landscapes 273
Laplace's equation 304 305
Law of Averages 23—24
Lebesgue density theorem 77 77 93
Lebesgue measure 13 16 112 192 264
Lebesgue measure, n-dimensional 13 17 28 112 143 266
Legendre spectrum 282
Legendre transform 281 282 287
Length 13 81
Length, scaling of 29
Level set 266 275
Level-k interval 35 62 152
Level-k set 127
Levy process 267 271—273
Levy stable process 271 271—273
Liapounov exponents 208—211 209 210 212
Lim sup sets 113
LIMIT 8—9
Limit of sequence 5
Limit, lower 9
Limit, upper 9
Line segment, dimension print 56
Line segment, uniform mass distribution on 14
Line set 176 176—179
Linear transformation 8
Lipschitz equivalence 236
Lipschitz function 8 10 30 103
Lipschitz invariance 41 48 55 56
Lipschitz transformation 30 32 34
Local dimension 283
Local product 103 196 202
Local structure of fractals 76—89
Logarithmic density 41
Logarithms 10
| Logistic map 189—193 191 192 212
Long range dependence 311
Loop 223 224—225
Loop, closed 201
Lorenz attractor 203—204 204
Lorenz equations 202—203
Lower box(-counting) dimension 41
Lower coarse multifractal spectrum 280
Lower density 77 84
Lower limit 9
Lubrication theory 305
Mandelbrot set 223 223—227 224 230 233 235
Mandelbrot, Benoit xxii xxv
Mapping(s) 6 8
Markov partition 189
Martingale 248 311
Mass distribution 11 12 277
Mass distribution and distribution of digits 151—152
Mass distribution and product rule 99 101
Mass distribution principle 60 60—61 131 200
Mass distribution, construction by repeated subdivision 14—15 15
Mass distribution, uniform, on line segment 14
Maximum modulus theorem 231
Maximum range 160
Maxwell's equations 310
Mean 21
Mean-value theorem 10 137 190
Measure(s) 11 11—17
Measure(s) on a set 11
Measure(s),  -finite 95
Measure(s), counting 13
Measure(s), Hausdorff 14 27—30 28 53 99 113
Measure(s), Hausdorff dimension of 209 288
Measure(s), invariant 208 208—211
Measure(s), Lebesgue 13 16 28 112 192
Measure(s), multifractal 211 277—296
Measure(s), n-dimensional Lebesgue 13 17 112 143
Measure(s), net 36 68
Measure(s), packing 50—53 51 88
Measure(s), probability 19
Measure(s), restriction of 14
Measure(s), self-similar 278 279 280 286
Measure(s), tangent 89
Method of moments, for multifractal spectrum 280—281 283
Metric dimension 41
Middle  Cantor set 64
Middle third Cantor set xvii xviii
Middle third Cantor set and repellers 188 189
Middle third Cantor set and self-similarity 123 124 129
Middle third Cantor set and tangents 87
Middle third Cantor set as attractor 189
Middle third Cantor set in intersections 112
Middle third Cantor set, box(-counting) dimension 47
Middle third Cantor set, construction of xviii 127
Middle third Cantor set, features xviii 123
Middle third Cantor set, generalization of 63—64
Middle third Cantor set, Hausdorff dimension 34—35 60 61
Middle third Cantor set, product 99 100
Minkowski content 45
Minkowski( — Bouligand) dimension 46
Modified box(-counting) dimension 49 49—50
Modified box(-counting) dimension, upper, and packing dimension 51—52
Modified von Koch curve 132 133—134
Moment sum 280 281
Monotonicity of box(-)counting dimension 48
Monotonicity of dimension print 55
Monotonicity of Hausdorff dimension 32 41
Monotonicity of packing dimension 51
Montel's theorem 218—219 221
Moser's twist theorem 207
Mountain skyline 271 298
Multifractal spectrum 277—286 283 289 292—294 315
Multifractal time 312 313 314
Multifractal(s) 211 277—296
Multifractal(s), coarse analysis 277 278—283
Multifractal(s), fine analysis 277 283—286
Multifractal(s), self-similar 286—296
Multifractional Brownian motion 271
Multiple points 265 275
Multivariate normal variable(s) 267
Natural fractals xxvi 146 147 298—300
Natural logarithms 10
Navier — Stokes equations 203 307 309
Neighbourhood 4 5 45 see
Net measure 36 68
Neural networks 277
Newton's method 237—241
Non-linear Cantor set 136—137 154
Non-removable set 180 181
Non-singular transformation 8
Normal distribution 23 260
Normal family 218 218—219
Normal family at a point 218
Normal numbers 151
Number theory 151—158
Often 110
One-sided dimension 54
One-to-one correspondence 7
One-to-one function 7
Onto function 7
Open ball 4
Open interval 4 138
Open set 5 32 41 187
Open set condition 129 130—134 249
Orbit 186 189 216 228
Orthogonal projection xxv 34 90—97 91 176 177
Packing dimension 50—53 51 284
Packing dimension and modified upper box dimension 51—52
Packing measure 50—53 51 88
Packing measure and Hausdorff measure 53
Parallel body 4
Parseval's theorem 73
Partial quotient 153
Percolation 251—255 252 253 254
Perfect set 220 242
Period 216
Period doubling 191—193
Period-p orbit 216 232 235
Period-p point 186 191
Periodic orbit 228 232
Periodic point 216 221
Phase transition 251
Physical applications xxvi 298—316
Pinch point 232
Plane cross section 202
Plant growth 300
Poincare section 202
Poincare — Bendixson theorem 201 202
Point mass 13
Pointwise convergence 10
Poisson's equation 307
Polynomials, Newton's method for 237—241
Population dynamics 186 190 193
Porous medium, flow through 305
Potential theoretic methods 70—72 92 111 248—249
Power spectrum 171 270
Prandtl number 203
Pre-fractal(s) 126 127
Pre-image 7
probability 18
Probability density function 23
Probability measure 18 19
Probability space 19
Probability theory 17—24
Probability, conditional 19
PRODUCT 99
Product formula 99—107
Product, cartesian 4 87 99 100
Projection theorems 90—93 180
Projection(s) 90—97 91
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