Fractal Geometry: Mathematical Foundations and Applications

400 Pages·2014·3.58 MB·English

by  Kenneth Falconer

Fractal Geometry

Fractal Geometry
Mathematical Foundations and Applications
Third Edition
Kenneth Falconer
UniversityofStAndrews,UK
Thiseditionfirstpublished2014
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Contents
Prefacetothefirstedition ix
Prefacetothesecondedition xiii
Prefacetothethirdedition xv
Coursesuggestions xvii
Introduction xix
PARTI FOUNDATIONS 1
1 Mathematicalbackground 3
1.1 Basicsettheory 3
1.2 Functionsandlimits 7
1.3 Measuresandmassdistributions 11
1.4 Notesonprobabilitytheory 17
1.5 Notesandreferences 24
Exercises 24
2 Box-countingdimension 27
2.1 Box-countingdimensions 27
2.2 Propertiesandproblemsofbox-countingdimension 34
*2.3 Modifiedbox-countingdimensions 38
2.4 Someotherdefinitionsofdimension 40
2.5 Notesandreferences 41
Exercises 42
3 Hausdorffandpackingmeasuresanddimensions 44
3.1 Hausdorffmeasure 44
3.2 Hausdorffdimension 47
3.3 CalculationofHausdorffdimension–simpleexamples 51
3.4 EquivalentdefinitionsofHausdorffdimension 53
vi CONTENTS
*3.5 Packingmeasureanddimensions 54
*3.6 Finerdefinitionsofdimension 57
*3.7 Dimensionprints 58
*3.8 Porosity 60
3.9 Notesandreferences 63
Exercises 64
4 Techniquesforcalculatingdimensions 66
4.1 Basicmethods 66
4.2 Subsetsoffinitemeasure 75
4.3 Potentialtheoreticmethods 77
*4.4 Fouriertransformmethods 80
4.5 Notesandreferences 81
Exercises 81
5 Localstructureoffractals 83
5.1 Densities 84
5.2 Structureof1-sets 87
5.3 Tangentstos-sets 92
5.4 Notesandreferences 96
Exercises 96
6 Projectionsoffractals 98
6.1 Projectionsofarbitrarysets 98
6.2 Projectionsofs-setsofintegraldimension 101
6.3 Projectionsofarbitrarysetsofintegraldimension 103
6.4 Notesandreferences 105
Exercises 106
7 Productsoffractals 108
7.1 Productformulae 108
7.2 Notesandreferences 116
Exercises 116
8 Intersectionsoffractals 118
8.1 Intersectionformulaeforfractals 119
*8.2 Setswithlargeintersection 122
8.3 Notesandreferences 128
Exercises 128
PARTII APPLICATIONSANDEXAMPLES 131
9 Iteratedfunctionsystems–self-similarandself-affinesets 133
9.1 Iteratedfunctionsystems 133
9.2 Dimensionsofself-similarsets 139
CONTENTS vii
9.3 Somevariations 143
9.4 Self-affinesets 149
9.5 Applicationstoencodingimages 155
*9.6 Zetafunctionsandcomplexdimensions 158
9.7 Notesandreferences 167
Exercises 167
10 Examplesfromnumbertheory 169
10.1 Distributionofdigitsofnumbers 169
10.2 Continuedfractions 171
10.3 Diophantineapproximation 172
10.4 Notesandreferences 176
Exercises 176
11 Graphsoffunctions 178
11.1 Dimensionsofgraphs 178
*11.2 Autocorrelationoffractalfunctions 188
11.3 Notesandreferences 192
Exercises 192
12 Examplesfrompuremathematics 195
12.1 DualityandtheKakeyaproblem 195
12.2 Vitushkin’sconjecture 198
12.3 Convexfunctions 200
12.4 Fractalgroupsandrings 201
12.5 Notesandreferences 204
Exercises 204
13 Dynamicalsystems 206
13.1 Repellersanditeratedfunctionsystems 208
13.2 Thelogisticmap 209
13.3 Stretchingandfoldingtransformations 213
13.4 Thesolenoid 217
13.5 Continuousdynamicalsystems 220
*13.6 Smalldivisortheory 225
*13.7 Lyapunovexponentsandentropies 228
13.8 Notesandreferences 231
Exercises 232
14 Iterationofcomplexfunctions–JuliasetsandtheMandelbrotset 235
14.1 GeneraltheoryofJuliasets 235
14.2 Quadraticfunctions–theMandelbrotset 243
14.3 Juliasetsofquadraticfunctions 248
14.4 Characterisationofquasi-circlesbydimension 256
14.5 Newton’smethodforsolvingpolynomialequations 258
viii CONTENTS
14.6 Notesandreferences 262
Exercises 262
15 Randomfractals 265
15.1 ArandomCantorset 266
15.2 Fractalpercolation 272
15.3 Notesandreferences 277
Exercises 277
16 BrownianmotionandBrowniansurfaces 279
16.1 Brownianmotioninℝ 279
16.2 Brownianmotioninℝn 285
16.3 FractionalBrownianmotion 289
16.4 FractionalBrowniansurfaces 294
16.5 Lévystableprocesses 296
16.6 Notesandreferences 299
Exercises 299
17 Multifractalmeasures 301
17.1 Coarsemultifractalanalysis 302
17.2 Finemultifractalanalysis 307
17.3 Self-similarmultifractals 310
17.4 Notesandreferences 320
Exercises 320
18 Physicalapplications 323
18.1 Fractalfingering 325
18.2 Singularitiesofelectrostaticandgravitationalpotentials 330
18.3 Fluiddynamicsandturbulence 332
18.4 Fractalantennas 334
18.5 Fractalsinfinance 336
18.6 Notesandreferences 340
Exercises 341
References 342
Index 357