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| Ïîèñê ïî óêàçàòåëÿì |
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| Falconer K.J. — Techniques in Fractal Geometry |
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| Ïðåäìåòíûé óêàçàòåëü |
 -field 7 129
(s-)energy 26
-neighbourhood 1 20
-neighbourhood, interior 226
-packing 22
-parallel body 1 20
Allen — Cahn equation 220
Almost all 10
Almost everywhere 10
Apollonian packing 55
Approximately self-similar set 62 67
Approximately self-similar set, arithmetic 117 122
Arithmetic-geometric mean inequality 16
Asymptotic eigenvalue distribution 222
atom 18
Attractor of dynamical system 207—223
Attractor of iterated function system 31
Average density 102—112 114 160—163
Average density, one-sided 112
Ball 1
Bi-Lipschitz equivalent 143 143—146
bi-Lipschitz mapping 21 24 143
Binary interval 139
Binary square 228
Borel measure 8
Borel regular 8
Borel set 2
Boundary 2
Bounded convergence theorem 12
Bounded distortion 62—69 65
Bounded set 2
Bounded variation 63
Box-Counting Dimension 19—22 20 51—56
Brownian motion 237—245
Cantor set, middle-third xi xiii 29 45 105 107 111 128 151 164
Chain rule 64
Characteristic exponent 101
Characteristic function 2
Closed ball 1
Closed set 2
Closure 2
Coarse multifractal spectrum 187
Coarse multifractal theory 186—192
Codomain 2
Compact 2
composition 2
Concave function 4
Conditional expectation 130
Conformal mapping 27 89
Convergence theorem 12
Convex function 4
Convolution 114
Cookie-cutter 56—69 108—110
Cookie-cutter set 62
Cookie-cutter system 59—69 62 71—88
Cookie-cutter system, dimension of 68 77 88
Cookie-cutter system, m-part 69
Correspondence, one-one 2
Correspondence, one-one, -cover 21
Covering number function 123
Cut-out set 51—56 136—142 147 224
Cut-out set, density 27 27—29 102—111 155—160
Cut-out set, lower 27 103
Cut-out set, upper 27 103
Density function 160
Density point 11 153
Diameter 1 19
Differential equation 207—246
Diffuse dimension distribution 181—183
DIMENSION 19—29
Dimension decomposition 177—183
Dimension derivative family 178—181
Dimension disintegration formula 178
Dimension measure 178 183
Dimension of attractor 207—223
Dimension of IFS attractor 36
Dimension of measure 169—176 170 171
Dimension, box-counting 19—22 20 51—56
Dimension, calculation of 24—27 41—57
Dimension, capacity 19
Dimension, entropy 19
Dimension, Hausdorff 21—23 22
Dimension, interior Minkowski 226 232
Dimension, local 25 169 169—183 185
Dimension, lower box-counting 20
Dimension, Minkowski 21 51
Dimension, packing 22—23 23
Dimension, pointwise 25 169 185
Dimension, pressure formula 75—79 77 88
Dimension, spectral 242
Dimension, upper box-counting 20
Dirichlet — Neumann bracketing 230
Disc 2
Distance (between sets) 1
Domain 2
Domain, fractal 236—245
Dominated Convergence Theorem 12
Eigenvalue counting function 223 223—230 243
Eigenvalues 222—230 236—245
Ellipsoid 208
entropy 84—88 85
Equivalent measures 11
Ergodic measure 83 97 102—103 175 175—176
Ergodic theorem 97—112 98
Ergodic theorem, approximate 100
Euclidean distance 1
Euclidean space 1
Exact dimensional 174 174—176
Expectation 129
Exterior product 214
Fatou's lemma 13
Fine multifractal spectrum 187
Fine multifractal theory 186—192
First variation equation 214
Flow 213
Fourier transform 120
Fractal geometry xi—xiv 19—40
Fractal, definition of xi
Frechet derivative 221
Fubini's theorem 13
Function 2
Functional attractor 220
Gap length 51
Gap-counting function 125
Gaussian kernel 231 240
Gibbs measure 71—75 75 82—84 87
Gibbs measure, multifractal analysis of 201—204
Graph-directed set 47 47—51 57 89—90
Hausdorff dimension 21—23 22
Hausdorff dimension of a measure 170 170—174
Hausdorff measure 21 24 36 77
Hausdorff metric 29
Hausdorff-type measure 191
Heat content 230
Heat equation 230—236 241—245
Henon attractor 212
Henon mapping 212
Hexakun 241—244
Hilbert transform 163 167 168
Histogram method 192
Hoelder continuous 89
HoElder exponent 25
Implicit methods 41—51
Indicator function 2
Infinitesimal vector 214
Injection 2
Integrable 12
integral 12
Integration 12—13
Interior 2
| Interior Minkowski dimension 226 232
Interior r-neighbourhood 226
interval 2
Invariant measure 38 79 79—84 97 102—103 175 175—176
Invariant set of dynamical system 207 209
Invariant set of iterated function system 31
Inverse 2
Irregular set 28
Iterate 2
Iterated function system/scheme (IFS) 29 29—39 60 89
Iterated function system/scheme (IFS), attractor of 31
Iterated function system/scheme (IFS), dimension of attractor 36
Iterated function system/scheme (IFS), graph-directed 47
Iterated function system/scheme (IFS), invariant set of 31
Iterated function system/scheme (IFS), probabilistic 36
Jensen's inequality 4
Julia set 90
Koch curve see von Koch curve
Laplacian 230 241 243
Laplacian, eigenvalues of 223—230 236—245
Lattice points 227
Lebesgue density theorem 27
Lebesgue integrable 13
Lebesgue measure 9 21 150
Legendre spectrum 191
Legendre transform 189 190 198 202
Liapounov exponent 101
LIMIT 3
Lipschitz constant 3
Lipschitz function 3 23 42—45
Lipschitz piece 28
Local dimension 25 169 169—176
Logarithmic density 19
Lorenz attractor 217—220
Lower box-counting dimension 20
Manifold 23
Mapping 2
Markov partition 91
Martingale 129—147 130
Martingale,  bounded 135
Martingale, convergence theorem 135
Mass distribution 6
Mass distribution principle 24
Measurable function 12
Measurable set 7
Measure 6—16 7
Measure preserving transformation 97
Measure, Borel 8
Measure, counting 9
Measure, finite 8
Measure, Hausdorff 21 21—23
Measure, Lebesgue 9
Measure, locally finite 8
Measure, packing 23 22—23
Measure, probability 8
Measure, restriction of 9
Mesh cube 20
Method of moments 192
Moment sums 189
Monotone Convergence Theorem 12
Monotonocity 23
Multifractal analysis 185—206
Multifractal analysis of Gibbs measures 201—204
Multifractal analysis of self-similar measures 192—201
Multifractal measure 185 185—206
Multifractal spectrum 185 185—206
Multifractal spectrum, coarse 187
Multifractal spectrum, fine 187
Multifractal spectrum, Hausdorff 187
Multifractal spectrum, packing 187
Navier — Stokes equation 223
Non-arithmetic 117 122
One-one mapping 2
Onto mapping 2
Open ball 1
Open set 2
Open set condition 35
Order-two density 104
Packing dimension 23
Packing dimension of a measure 170 170—174
Packing measure 22—23 23
Perron — Frobenius theorem 50
Point mass 9
Post-critically finite set 243
Potential 26
Pre-fractal 31 31—33
Pressure 71—79 75 82 87 202
Principle of not feeling the boundary 233
Probabilistic IFS 36 36—39
Probability measure 8
Pseudo-metric 15
Quasi-self-similar set 67
Random cut-out set 136—142 147
Random walk 237—245
RANGE 2
Reaction-diffusion equation 220—223
Rectifiability 27—29
Rectifiable 28 159 159—160 166
Regular set 28
Renewal equation 113 115—117 236
Renewal Theorem 113—128 117 122
Repeated subdivision 9 9—10
Repeller 59
Residence measure 176
Restriction of measure 9
Riemann zeta function 226
Ruelle — Perron — Frobenius theorem 80
Scaling property 22
Schauder fixed point theorem 81
Schrodinger equation 223
Self-affine measure 19
Self-affine set 33 35
Self-conformal set 35 90
Self-similar measure 39 102 172
Self-similar measure, multifractal analysis of 192—201
Self-similar set xiii 35 46 123—127 143—146
Self-similar set, approximately 62 67
Self-similar set, quasi- 67
Self-similar set, statistically 142
Self-similar set, sub- 46 46—48
Self-similar set, super- 46 46—48
Sequence notation 32
Shift invariance 154
Sierpinski carpet xi xiii
Sierpinski gasket see Sierpinski triangle
Sierpinski triangle xi xiii 32 57 236—245
Sierpinski triangle, extended 237
Similarity transformation 22
Simple function 12
Sinai-Bowen-Ruelle operator 79
Singular integral 110 163—167
Singular value 208
Singular value function 208
Singularity spectrum 185
Spectral dimension 242
Stable, countably 24
Stable, finitely 24
Statistically self-similar set 142
Strong separation condition 35
Sub-self-similar set 46 46—48
Subadditive sequence 3 85
Submartingale 130
Submultiplicative sequence 4
Super-self-similar set 46 46—48
Supermartingale 130 132—135
Supermartingale convergence theorem 134
Support of function 2
Support of measure 8
Surjection 2
Tangent measure 149—168 150
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