Further Developments in Fractals and Related Fields: Mathematical Foundations and Connections

!Cover......Page 1
Further Developments in \rFractals and Related Fields......Page 4
Preface......Page 8
Contents......Page 10
1 Introduction......Page 15
2.1 Background: Complexity and Sturmian Words......Page 17
2.2 Arnoux–Rauzy Words and Episturmian Words: Definition......Page 18
2.3 Ternary AR Words: Renormalization......Page 19
3 The Rauzy Gasket......Page 21
3.1 The Rauzy Gasket as an Iterated Function System......Page 22
3.2 Symbolic Dynamics for the Rauzy Gasket......Page 24
4.1 The Minkowski Question Mark Function......Page 26
4.2 The Sierpiński Gasket......Page 28
4.3 A Generalization of the Minkowski Question MarkFunction......Page 29
5 The Apollonian Gasket......Page 30
6.1 The Fully Subtractive Algorithm......Page 32
6.2 The Fully Subtractive Algorithm as an Extension of the Rauzy Gasket......Page 33
6.3 Two Properties of the Rauzy Gasket......Page 34
7 Final Remarks......Page 35
References......Page 36
1 Introduction......Page 38
2 Prevalence......Page 40
3 On the Graph of a Perturbed Fractional Brownian Motion......Page 41
4 Proof of Theorem 3......Page 44
5 The Case of α-Hölderian Functions......Page 45
References......Page 46
1 Introduction......Page 48
2 Three Families of Exponents of Approximation......Page 50
3 Approximation to Points in the Middle Third Cantor Set......Page 54
References......Page 57
1 Introduction......Page 59
2 The One-Dimensional Case......Page 61
4 Self-similar Sets and Singular Integrals......Page 63
5 Self-similar Sets in Heisenberg Groups......Page 64
6 Riesz-Type Kernels in Heisenberg Groups......Page 67
7 Δh-Removability and Singular Integrals......Page 68
8 Δh-Removable Self-similar Cantor Sets in Hn......Page 70
9 Concluding Comments......Page 71
References......Page 72
1 Introduction......Page 74
2 Relationships Between Davenport and Fourier Series......Page 76
3 Discontinuities of Davenport Series......Page 78
4 The Jump Operator......Page 80
5 Pointwise Hölder Regularity......Page 84
6 Sparse Davenport Series......Page 88
6.1 Sparse Sets and Link with Lacunary and HadamardSequences......Page 89
6.2 Decay of Sequences with Sparse Support and Behaviorof the Jump Operator......Page 90
6.3 Pointwise Regularity of Sparse Davenport Series......Page 92
7 Implications for Multifractal Analysis......Page 94
8 Convergence and Global Regularity of Davenport Series......Page 96
8.1 Preliminaries on Multivariate Arithmetic Functions......Page 97
8.2 Davenport Expansions Versus Fourier Expansions......Page 98
8.3 Regularity of the Sum of a Davenport Series......Page 99
9.1 Optimality of Lemma 2......Page 104
9.2 Hecke\'s Functions......Page 106
9.3 Spectrum of Singularities of Compensated Pure JumpsFunctions......Page 107
9.4 p-Exponent......Page 108
9.5 Directional Regularity......Page 109
10 Proof of Theorem 1......Page 110
11.1 Locations of the Singularities......Page 114
11.2 Size and Large Intersection Properties of the Sets La(α),Connection with the Duffin–Schaeffer and CatlinConjectures......Page 115
11.3 End of the Proof......Page 121
References......Page 123
1 Introduction......Page 125
1.1 Basic Definitions......Page 126
2.1 Cutting up Ellipses......Page 128
2.2 The Affinity Dimension......Page 131
2.3 Generic Results......Page 132
2.4 Sets with Dimension Attaining the Affinity Dimension......Page 134
3.1 Bedford–McMullen Carpets......Page 136
3.3 Box-Like Sets......Page 137
4 Self-affine Functions......Page 139
5.1 Multifractal Analysis of Measures on Self-affine Sets......Page 140
5.2 Nonlinear Analogues......Page 141
References......Page 142
The Multifractal Spectra of V-Statistics......Page 145
1 Introduction......Page 146
2 V-Statistics......Page 148
3 Topological Entropy......Page 149
4 Proof of Theorem 1......Page 151
5 Example: Shift Dynamics......Page 153
References......Page 161
1 Introduction......Page 162
2 Projections of Measures Invariant Under the Geodesic Flow......Page 164
3 Quantum Unique Ergodicity......Page 165
References......Page 168
1 Fractal Tubes......Page 170
2.1 Multifractal Spectra......Page 173
2.3 The Multifractal Formalism......Page 175
3.1 Multifractal Tubes......Page 176
3.2 Multifractal Tubes of Self-similar Measures......Page 178
3.3 How Does One Prove Theorem 2 on the AsymptoticBehaviour of Multifractal Tubes of Self-similarMeasures?......Page 183
4.1 Multifractal Tube Measures......Page 191
4.2 Multifractal Tube Measures of Self-similar Measures......Page 193
References......Page 199
1 Introduction......Page 201
2 Preliminaries and the Scheme of the Proof......Page 202
3 Lower Estimates of Hausdorff Measure......Page 205
4 Upper Bound for Hausdorff Measure......Page 212
References......Page 220
1 Introduction......Page 221
2 Preliminaries......Page 223
3 The Law of Iterated Logarithm: Abstract Setting......Page 227
4 The Law of Iterated Logarithm: Meromorphic Functions......Page 233
5 Equilibrium States Versus Hausdorff Measures......Page 238
References......Page 241
1 Introduction......Page 243
1.1 Cookie-Cutter and Cookie-Cutter-Like Constructions......Page 244
1.2 Graph-Directed Construction......Page 245
1.3 Cookie-Cutter-Like Sets with Graph-Directed Construction (GCCL)......Page 246
2.1 Basic Assumption......Page 248
3 Four Properties of GCCL......Page 249
3.2 Proofs of Four Properties......Page 250
4.1 Proof of Theorem 1......Page 254
4.2 Proof of Theorem 2......Page 255
4.3 Proof of Theorem 3......Page 258
References......Page 261
1 Introduction......Page 263
2 Gaussian Random Fields......Page 264
2.1 Space-Anisotropic Gaussian Random Fields......Page 265
2.2 Time-Anisotropic Gaussian Random Fields......Page 266
2.3 Assumptions......Page 267
3 Analytic Results......Page 269
3.1 Exact Modulus of Continuity and LIL......Page 270
3.2 Chung\'s LIL and Modulus of Nondifferentiability......Page 272
3.3 Regularity of Local Times......Page 274
4 Fractal Properties......Page 277
4.1 Hausdorff Dimension Results......Page 278
4.2 The Fourier Dimension and Salem Sets......Page 280
4.3 Packing Dimension Results......Page 281
4.4 Uniform Dimension Results......Page 283
4.5 Exact Hausdorff Measure Functions......Page 285
4.6 Exact Packing Measure Functions......Page 287
4.7 Hitting Probabilities and Intersections of Gaussian Random Fields......Page 288
References......Page 291