Recent Developments in Fractals and Related Fields (eBook)

ANHA Series Preface 
7 
Preface 
10 
Contents 
12 
List of Contributors 
15 
Part I Geometric Measure Theory and Multifractals 19
Occupation Measure and Level Sets of the Weierstrass–Cellerier Function 20
1 Introduction 20
2 Notation and Preliminary Results 22
3 Main Results 24
4 Proof of Lemma 3 28
References 34
Space-Filling Functions and Davenport Series 36
1 Introduction 36
2 Definitions 36
2.1 Hölder Spaces and Hölder Exponents 37
2.2 Uniform Hölder Spaces and Strongly Monohölder Functions 38
3 The Peano Function 39
4 A Strong Monohölderianity Criterion 41
5 The Lebesgue Function 42
6 The Schoenberg Function 44
7 The Cantor Function 46
8 Hölder Exponent of p-adic Davenport Series 49
References 51
Dimensions and Porosities 52
1 Porous Sets 52
2 Porous Measures 54
3 Mean Porous Measures 56
References 59
On Upper Conical Density Results 61
1 Introduction 61
2 Notation and Preliminaries 61
3 Packing Type Measures 63
4 Measures with Positive Dimension 66
5 Purely Unrectifiable Measures 67
References 69
On the Dimension of Iterated Sumsets 71
1 Introduction and Statement of Results 71
2 Examples and Proof of the Main Result 73
2.1 Basic Facts 73
2.2 Examples for Hausdorff Dimension 74
2.3 Examples for Hausdorff and Box-Counting Dimensions 78
2.4 Proof of the Main Result 85
3 Plünnecke Estimates for Box-Counting Dimensions 86
References 88
Geometric Measures for Fractals 89
1 Introduction 89
2 Minkowski Content 90
3 Sequences of Signed Measures 92
4 Curvature Measures and Fractal Curvatures 94
4.1 Curvature Measures 94
4.2 Fractal Curvature Measures 96
5 Curvature Measures for Self-Similar Sets 97
6 Proof of Theorem 5.1 101
7 Generalizations 103
References 104
Part II Harmonic and Functional Analysis and, Signal Processing 106
A Walk from Multifractal Analysis to Functional Analysis with S Spaces, and Back 107
1 Introduction 107
1.1 Where do the Spaces S Come from? 107
1.2 Heuristic Description 108
1.3 Outline of this Chapter 108
2 S Metrizable Topology 109
2.1 Definitions and Notation 109
2.2 More Functional Analysis: p-Convexity 110
2.3 More Functional Analysis: Typical Properties 111
Case p0 = 1 111
Case p0< 1
2.4 More Functional Analysis: Properties of the Dual 114
3 Contributions to Multifractal Analysis 115
3.1 Prevalent Properties 115
3.2 Quasi-Sure Properties 116
3.3 On the Maximal Spectrum in Locally p-Convex Spaces 117
3.4 Open Questions 119
References 120
Concentration of the Integral Norm of Idempotents 121
1 Introduction and Statement of Results 121
2 Uniform p-Concentration 125
3 Failure of Uniform 1-Concentration on Zq 130
4 The 2-Concentration on Measurable Sets 131
5 Improvement of Constants for p not an Even Integer 136
References 142
Le calcul symbolique dans certaines algèbres de type Sobolev 144
1 Généralités sur le calcul symbolique 144
1.1 Le cas des algèbres de fonctions 145
1.2 Le cas des algèbres régulières 145
1.3 Vers un calcul symbolique maximal? 147
1.4 Le cas des algèbres sur le cercle 147
2 Calcul symbolique dans les espaces de Sobolev 148
2.1 Définition et théorème principal 148
2.2 Reformulation du théorème principal 149
2.3 Preuve des corollaires 7 et 8 150
2.4 Preuve du théorème 2.3 150
Préliminaires 150
Normes équivalentes dans l'espace de Sobolev 151
La 2-variation d'une fonction 151
Les détails de la preuve 152
3 Diverses Extensions 153
3.1 Le cas des espaces de Besov et de Lizorkin-Triebel 153
3.2 Le cas des espaces définis sur Rn 153
3.3 Le cas des espaces de Sobolev à valeurs vectorielles 155
4 Régularité du calcul symbolique 155
4.1 Calcul symbolique borné 155
4.2 Calcul symbolique de classe Cr 156
References 156
Lp-Norms and Fractal Dimensions of Continuous Function Graphs 158
1 Introduction 158
2 Operator Norms 161
3 Application to Box Dimension 166
4 Application to Several Fractal Indices 168
5 Examples 172
5.1 Riemann Functions 172
5.2 Weierstrass Functions 174
5.3 Fractal Sums of Pulses 174
References 176
Uncertainty Principles, Prolate Spheroidal Wave Functions, and Applications 178
1 Introduction 178
2 Uncertainty Principles and Prolate Spheroidal Wave Functions 180
2.1 Uncertainty Principles 180
2.2 Properties of the PSWFs and Their Eigenvalues 182
3 Computation of the PSWFs and Their Eigenvalues 184
3.1 Mathematical Preliminaries 184
3.2 A Classical Computational Method 186
3.3 Matrix Representation of Qc and PSWFs 187
3.4 A Quadrature Method for the Computation of the PSWFs 190
3.5 Examples 195
4 Computation of the Spectrum of High Frequency PSWFs 197
5 Applications of the PSWFs 199
5.1 PSWFs and Quality of Approximation 199
5.2 Exact Reconstruction of Band-Limited Functions by the PSWFs 200
5.3 Examples 201
References 202
2-Microlocal Besov Spaces 204
1 Introduction and Preliminaries 204
2 Characterization with Wavelets 208
3 The Local Spaces Bs,s'p,q(U)loc 210
3.1 Definition and Wavelet Characterization 210
3.2 Embeddings 211
3.3 The 2-Microlocal Domain 212
Acknowledgment 213
References 213
Refraction on Multilayers 215
1 Precise Measurement 215
2 Fuzzy Measurements and Algebra 216
References 218
Wavelet Shrinkage: From Sparsity and Robust Testing to Smooth Adaptation 219
1 Introduction 219
2 WaveShrink: Estimation by Sparse Transform and Thresholding 222
2.1 Background 222
2.2 WaveShrink by Soft Thresholding 223
3 From Non-Parametric Statistical Decision to Sparsity 224
3.1 Motivation 224
3.2 Detection of Random Signals with Unknown Distribution and Prior in White Gaussian Noise 225
3.3 Sparse Sequences and Detection Thresholds 227
3.4 Application to WaveShrink by Soft Thresholding and the Universal Detection Threshold 228
4 Smooth Adapted WaveShrink with Adapted Detection Thresholds 230
4.1 SSBS Functions 231
4.2 Adapted Detection Thresholds 235
4.3 Experimental Results 236
5 Conclusion 240
References 243
Part III Dynamical Systems and Analysis on Fractals 245
Simple Infinitely Ramified Self-Similar Sets 246
1 Introduction 246
1.1 A Few Remarks on Fractal Analysis 246
1.2 The Advantage of Self-Similarity 247
1.3 Contents of the Chapter 249
2 Self-Similar Sets of Low Complexity 249
2.1 Basic Definitions 249
2.2 Counting Neighbor Types to Measure Complexity 250
3 Symmetric Examples 252
3.1 Carpet and Gasket Constructions 252
3.2 Fractal m-gons 253
4 The Boundary Structure 255
4.1 Self-Similarity of the Intersection Sets 255
4.2 Intersection Sets as Minimal Cuts 256
5 Examples with Exact Overlap of Pieces 258
References 259
Quantitative Uniform Hitting in Exponentially Mixing Systems 261
1 Introduction 261
2 Probabilistic Setting 263
3 On the Exponentially Mixing Property 264
4 Weighted Borel–Cantelli Lemma 265
5 Fundamental Inequalities 266
5.1 Basic Inequalities 266
6 Proofs of Theorems 271
6.1 Proof of Theorem 2.1 271
6.2 Proof of Theorem 1.1 272
7 Gibbs Measures on Subshifts of Finite Type 272
References 275
Some Remarks on the Hausdorff and Spectral Dimension of V-Variable Nested Fractals 277
1 Introduction 277
2 The Model: Two Shapes and Their Random Mixing 278
2.1 Classical Models 278
2.2 The V-Variable Model 280
3 Hausdorff Dimension 281
4 Spectral Dimension 283
4.1 The Deterministic Case 283
4.2 The V-Variable Case 284
4.3 Construction of the Energy Form 284
4.4 Spectral Asymptotics 286
5 Generalization: V-Variable Nested Fractals 288
References 291
Cantor Boundary Behavior of Analytic Functions 293
1 Introduction 293
2 The Basic Setup 294
3 The Cantor Boundary Behavior 295
4 The Complex Weierstrass Functions 298
5 Cauchy Transform on Sierpinski Gasket 300
References 304
Measures of Full Dimension on Self-Affine Graphs 305
1 Introduction 305
2 Notation and Background 308
3 Classical Self-Affine Graphs and Their Sofic Coding 310
3.1 McMullen–Bedford (1984) 310
3.2 Przytycki and Urbanski (1989) and Urbanski (1990) 311
3.3 Kamae (1986) 314
References 317
Part IV Stochastic Processes and Random Fractals 319
A Process Very Similar to Multifractional Brownian Motion 320
1 Introduction and Statement of the Main Results 320
2 The Main Ideas of the Proofs 325
3 Some Technical Lemmas 332
References 334
Gaussian Fields Satisfying Simultaneous Operator Scaling Relations 336
1 Introduction 336
2 A First Example of a Field Satisfying Simultaneous Operator Scaling Properties 339
3 A General Approach 341
4 Existence Results 343
4.1 Real Diagonalizable Part of a Matrix 343
4.2 Assumptions on Matrices E1,…,Em 343
5 Reformulation of the Problem in Terms of Group Self-Similarity 344
6 Definition of the Desired Gaussian Field 346
6.1 Definition of a Suitable Spectral Density 346
6.2 Definition of Renormalization Directions 347
6.3 Proof of Theorem 1 349
References 350
On Randomly Placed Arcs on the Circle 351
1 Introduction and Statement of the Results 351
1.1 Size Properties of the Set E 353
1.2 Large Intersection Properties of the Set E 354
2 Proof of Theorems 1 and 2 356
3 Proof of Corollary 1 357
4 Proof of Corollary 2 357
5 Proof of Proposition 2 358
References 358
T-Martingales, Size Biasing, and Tree Polymer Cascades 360
1 Introduction 360
2 Background and Notation 361
2.1 Some Special Notation and Assumptions 363
3 T-Martingales and Size-Bias Theory 364
4 Asymptotic Polymer Path Free Energy-Type Calculations for Weak and Strong Disorder 367
5 Diffusive Limits Under Full Range of Weak Disorder 373
6 Vector Cascades 378
7 Related Directions in T-Martingale Theory 385
Acknowledgments 386
References 386
Part V Combinatorics on Words 388
Univoque Numbers and Automatic Sequences 389
1 Introduction 389
2 A Class of Sequences Belonging to strict 390
3 Proof of Theorem 1 392
3.1 Proof of the First Assertion 392
3.2 The Sequence limk r(k)((u 0)) is Not Periodic 393
4 Automatic Sequences and the Sets and strict 393
5 Alphabets with More Than Two Letters 395
6 Conclusion 396
References 396
A Crash Look into Applications of Aperiodic Substitutive Sequences 398
1 Into the Past: The ``Gang of Five" 398
2 Non-Bragg Diffraction in Systems with Aperiodic Order and Lebesgue Classification 399
3 Substitutive Sequences as Sources for Discrete Laplacians: Spectrum, Band Structure, and Eigenstates in Localization Studies 401
References 402
Invertible Substitutions with a Common Periodic Point 405
1 Introduction 405
2 Some Known Results Concerning Invertible Substitutions 406
2.1 Frequency of a Substitution, the Generating Matrix 406
2.2 Sturmian Words 407
3 Rauzy Fractals of Invertible Substitutions 408
4 Characterization of Rauzy Fractals Stepped Surface 409
4.1 The Stepped Surface 409
4.2 A Tiling Associated with the Stepped Surface 409
4.3 Invertible Substitutions with a Given Incidence Matrix 410
5 Proof of Theorem 1 411
References 412
Some Studies on Markov-Type Equations 414
1 Introduction 414
2 Proof of Theorem 1 418
3 The Structure of Solutions of Equation (1) 421
References 422