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Falconer K.J. Ч Techniques in Fractal Geometry
Falconer K.J. Ч Techniques in Fractal Geometry

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Ќазвание: Techniques in Fractal Geometry

јвтор: Falconer K.J.

јннотаци€:

This book addressees a variety of techniques and applications in fractal geometry. It examines such topics as implicit methods and the theory of dimensions of measures, the thermodynamic formalism, the tangent of space method and the ergodic theorem. Each chapter ends with brief notes on the development and current state of the subject. Provides a clear guide to applications and recent trends in fractal geometry. There are numerous diagrams and illustrative examples.


язык: en

–убрика: ћатематика/ћатематическа€ ‘изика/

—татус предметного указател€: √отов указатель с номерами страниц

ed2k: ed2k stats

√од издани€: 1997

 оличество страниц: 256

ƒобавлена в каталог: 16.04.2005

ќперации: ѕоложить на полку | —копировать ссылку дл€ форума | —копировать ID
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ѕредметный указатель
$\sigma$-field      7 129
(s-)energy      26
-neighbourhood      1 20
-neighbourhood, interior      226
-packing      22
-parallel body      1 20
Allen Ч 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      207Ч223
Attractor of iterated function system      31
Average density      102Ч112 114 160Ч163
Average density, one-sided      112
Ball      1
Bi-Lipschitz equivalent      143 143Ч146
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      62Ч69 65
Bounded set      2
Bounded variation      63
Box-Counting Dimension      19Ч22 20 51Ч56
Brownian motion      237Ч245
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      186Ч192
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      56Ч69 108Ч110
Cookie-cutter set      62
Cookie-cutter system      59Ч69 62 71Ч88
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      51Ч56 136Ч142 147 224
Cut-out set, density      27 27Ч29 102Ч111 155Ч160
Cut-out set, lower      27 103
Cut-out set, upper      27 103
Density function      160
Density point      11 153
Diameter      1 19
Differential equation      207Ч246
Diffuse dimension distribution      181Ч183
DIMENSION      19Ч29
Dimension decomposition      177Ч183
Dimension derivative family      178Ч181
Dimension disintegration formula      178
Dimension measure      178 183
Dimension of attractor      207Ч223
Dimension of IFS attractor      36
Dimension of measure      169Ч176 170 171
Dimension, box-counting      19Ч22 20 51Ч56
Dimension, calculation of      24Ч27 41Ч57
Dimension, capacity      19
Dimension, entropy      19
Dimension, Hausdorff      21Ч23 22
Dimension, interior Minkowski      226 232
Dimension, local      25 169 169Ч183 185
Dimension, lower box-counting      20
Dimension, Minkowski      21 51
Dimension, packing      22Ч23 23
Dimension, pointwise      25 169 185
Dimension, pressure formula      75Ч79 77 88
Dimension, spectral      242
Dimension, upper box-counting      20
Dirichlet Ч Neumann bracketing      230
Disc      2
Distance (between sets)      1
Domain      2
Domain, fractal      236Ч245
Dominated Convergence Theorem      12
Eigenvalue counting function      223 223Ч230 243
Eigenvalues      222Ч230 236Ч245
Ellipsoid      208
entropy      84Ч88 85
Equivalent measures      11
Ergodic measure      83 97 102Ч103 175 175Ч176
Ergodic theorem      97Ч112 98
Ergodic theorem, approximate      100
Euclidean distance      1
Euclidean space      1
Exact dimensional      174 174Ч176
Expectation      129
Exterior product      214
Fatou's lemma      13
Fine multifractal spectrum      187
Fine multifractal theory      186Ч192
First variation equation      214
Flow      213
Fourier transform      120
Fractal geometry      xiЧxiv 19Ч40
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      71Ч75 75 82Ч84 87
Gibbs measure, multifractal analysis of      201Ч204
Graph-directed set      47 47Ч51 57 89Ч90
Hausdorff dimension      21Ч23 22
Hausdorff dimension of a measure      170 170Ч174
Hausdorff measure      21 24 36 77
Hausdorff metric      29
Hausdorff-type measure      191
Heat content      230
Heat equation      230Ч236 241Ч245
Henon attractor      212
Henon mapping      212
Hexakun      241Ч244
Hilbert transform      163 167 168
Histogram method      192
Hoelder continuous      89
HoElder exponent      25
Implicit methods      41Ч51
Indicator function      2
Infinitesimal vector      214
Injection      2
Integrable      12
integral      12
Integration      12Ч13
Interior      2
Interior Minkowski dimension      226 232
Interior r-neighbourhood      226
interval      2
Invariant measure      38 79 79Ч84 97 102Ч103 175 175Ч176
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 29Ч39 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      223Ч230 236Ч245
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 42Ч45
Lipschitz piece      28
Local dimension      25 169 169Ч176
Logarithmic density      19
Lorenz attractor      217Ч220
Lower box-counting dimension      20
Manifold      23
Mapping      2
Markov partition      91
Martingale      129Ч147 130
Martingale, $L^2$ bounded      135
Martingale, convergence theorem      135
Mass distribution      6
Mass distribution principle      24
Measurable function      12
Measurable set      7
Measure      6Ч16 7
Measure preserving transformation      97
Measure, Borel      8
Measure, counting      9
Measure, finite      8
Measure, Hausdorff      21 21Ч23
Measure, Lebesgue      9
Measure, locally finite      8
Measure, packing      23 22Ч23
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      185Ч206
Multifractal analysis of Gibbs measures      201Ч204
Multifractal analysis of self-similar measures      192Ч201
Multifractal measure      185 185Ч206
Multifractal spectrum      185 185Ч206
Multifractal spectrum, coarse      187
Multifractal spectrum, fine      187
Multifractal spectrum, Hausdorff      187
Multifractal spectrum, packing      187
Navier Ч 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 170Ч174
Packing measure      22Ч23 23
Perron Ч Frobenius theorem      50
Point mass      9
Post-critically finite set      243
Potential      26
Pre-fractal      31 31Ч33
Pressure      71Ч79 75 82 87 202
Principle of not feeling the boundary      233
Probabilistic IFS      36 36Ч39
Probability measure      8
Pseudo-metric      15
Quasi-self-similar set      67
Random cut-out set      136Ч142 147
Random walk      237Ч245
RANGE      2
Reaction-diffusion equation      220Ч223
Rectifiability      27Ч29
Rectifiable      28 159 159Ч160 166
Regular set      28
Renewal equation      113 115Ч117 236
Renewal Theorem      113Ч128 117 122
Repeated subdivision      9 9Ч10
Repeller      59
Residence measure      176
Restriction of measure      9
Riemann zeta function      226
Ruelle Ч Perron Ч 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      192Ч201
Self-similar set      xiii 35 46 123Ч127 143Ч146
Self-similar set, approximately      62 67
Self-similar set, quasi-      67
Self-similar set, statistically      142
Self-similar set, sub-      46 46Ч48
Self-similar set, super-      46 46Ч48
Sequence notation      32
Shift invariance      154
Sierpinski carpet      xi xiii
Sierpinski gasket      see Sierpinski triangle
Sierpinski triangle      xi xiii 32 57 236Ч245
Sierpinski triangle, extended      237
Similarity transformation      22
Simple function      12
Sinai-Bowen-Ruelle operator      79
Singular integral      110 163Ч167
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 46Ч48
Subadditive sequence      3 85
Submartingale      130
Submultiplicative sequence      4
Super-self-similar set      46 46Ч48
Supermartingale      130 132Ч135
Supermartingale convergence theorem      134
Support of function      2
Support of measure      8
Surjection      2
Tangent measure      149Ч168 150
1 2
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