Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings

Lapidus......Page 1
Lapidus0......Page 2
Fractal Geometry, Complex Dimensions and Zeta Functions......Page 4
Overview......Page 8
Preface......Page 12
Contents......Page 16
List of Figures......Page 22
List of Tables......Page 26
Introduction......Page 28
1.1 The Geometry of a Fractal String......Page 36
1.1.1 The Multiplicity of the Lengths......Page 39
1.1.2 Example: The Cantor String......Page 40
1.2 The Geometric Zeta Function of a Fractal String......Page 43
1.2.1 The Screen and the Window......Page 46
1.2.2 The Cantor String (continued)......Page 49
1.3 The Frequencies of a Fractal String and the Spectral Zeta Function......Page 51
1.4 Higher-Dimensional Analogue: Fractal Sprays......Page 54
1.5 Notes......Page 57
2.1 Construction of a Self-Similar Fractal String......Page 60
2.1.1 Relation with Self-Similar Sets......Page 62
2.2 The Geometric Zeta Function of a Self-Similar String......Page 65
2.2.1 Self-Similar Strings with a Single Gap......Page 67
2.3.1 The Cantor String......Page 68
2.3.2 The Fibonacci String......Page 70
2.3.3 The Modified Cantor and Fibonacci Strings......Page 73
2.3.4 A String with Multiple Poles......Page 74
The Golden String......Page 76
2.4 The Lattice and Nonlattice Case......Page 79
2.5 The Structure of the Complex Dimensions......Page 81
2.6 The Asymptotic Density of the Poles in the Nonlattice Case......Page 88
2.7 Notes......Page 89
3 Complex Dimensions of Nonlattice Self-Similar Strings: Quasiperiodic Patterns and Diophantine Approximation......Page 91
3.1 Dirichlet Polynomial Equations......Page 92
3.1.1 The Generic Nonlattice Case......Page 93
3.2.1 Generic and Nongeneric Nonlattice Equations......Page 94
3.2.2 The Complex Roots of the Golden Plus Equation......Page 99
3.3 The Structure of the Complex Roots......Page 101
3.4 Approximating a Nonlattice Equation by Lattice Equations......Page 106
3.4.1 Diophantine Approximation......Page 109
3.4.2 The Quasiperiodic Pattern of the Complex Dimensions......Page 112
3.4.3 Application to Nonlattice Strings......Page 115
3.5 Complex Roots of a Nonlattice Dirichlet Polynomial......Page 118
3.5.1 Continued Fractions......Page 119
3.5.2 Two Generators......Page 120
3.5.3 More than Two Generators......Page 126
3.6 Dimension-Free Regions......Page 129
3.7 The Dimensions of Fractality of a Nonlattice String......Page 136
3.7.1 The Density of the Real Parts......Page 137
3.8 A Note on the Computations......Page 141
3.9 Notes......Page 143
4 Generalized Fractal Strings Viewed as Measures......Page 144
4.1 Generalized Fractal Strings......Page 145
4.1.1 Examples of Generalized Fractal Strings......Page 147
4.2 The Frequencies of a Generalized Fractal String......Page 149
4.2.1 Completion of the Harmonic String: Euler Product......Page 152
4.3 Generalized Fractal Sprays......Page 154
4.4 The Measure of a Self-Similar String......Page 155
4.4.1 Measures with a Self-Similarity Property......Page 157
4.5 Notes......Page 160
5.1 Introduction......Page 161
5.1.1 Outline of the Proof......Page 163
5.1.2 Examples......Page 164
5.2 Preliminaries: The Heaviside Function......Page 166
5.3 Pointwise Explicit Formulas......Page 170
5.3.1 The Order of Growth of the Sum over the Complex Dimensions......Page 181
5.4 Distributional Explicit Formulas......Page 182
5.4.1 Extension to More General Test Functions......Page 187
5.4.2 The Order of the Distributional Error Term......Page 192
5.5 Example: The Prime Number Theorem......Page 198
5.5.1 The Riemann–von Mangoldt Formula......Page 200
5.6 Notes......Page 201
6 The Geometry and the Spectrum of Fractal Strings......Page 203
6.1.1 The Geometric Local Terms......Page 204
6.1.2 The Spectral Local Terms......Page 206
6.1.4 The Distribution xω log mx......Page 207
6.2.1 The Geometric Counting Function of a Fractal String......Page 208
6.2.2 The Spectral Counting Function of a Fractal String......Page 209
6.2.3 The Geometric and Spectral Partition Functions......Page 211
6.3.1 The Density of Geometric and Spectral States......Page 213
6.3.2 The Spectral Operator and its Euler Product......Page 214
6.4 Self-Similar Strings......Page 218
6.4.1 Lattice Strings......Page 219
6.4.2 Nonlattice Strings......Page 222
Lattice Case......Page 224
Nonlattice Case......Page 226
6.5.1 The a-String......Page 227
6.5.2 The Spectrum of the Harmonic String......Page 230
6.6 Fractal Sprays......Page 231
6.6.1 The Sierpinski Drum......Page 232
6.6.2 The Spectrum of a Self-Similar Spray......Page 236
7 Periodic Orbits of Self-Similar Flows......Page 237
7.1 Suspended Flows......Page 238
7.1.1 The Zeta Function of a Dynamical System......Page 239
7.2 Periodic Orbits, Euler Product......Page 241
7.3 Self-Similar Flows......Page 243
7.3.1 Examples of Self-Similar Flows......Page 247
7.3.2 The Lattice and Nonlattice Case......Page 248
7.4 The Prime Orbit Theorem for Suspended Flows......Page 250
7.4.2 Lattice Flows......Page 252
7.4.3 Nonlattice Flows......Page 253
7.5.1 Two Generators......Page 254
7.5.2 More Than Two Generators......Page 256
7.6 Notes......Page 259
8 Fractal Tube Formulas......Page 260
8.1 Explicit Formulas for the Volume of Tubular Neighborhoods......Page 261
8.1.1 The Pointwise Tube Formula......Page 266
8.1.2 Example: The a-String......Page 269
8.2 Analogy with Riemannian Geometry......Page 270
8.3 Minkowski Measurability and Complex Dimensions......Page 271
8.4.1 Generalized Cantor Strings......Page 276
8.4.2 Lattice Self-Similar Strings......Page 279
8.4.3 The Average Minkowski Content......Page 284
8.4.4 Nonlattice Self-Similar Strings......Page 287
8.5 Notes......Page 291
9 Riemann Hypothesis and Inverse Spectral Problems......Page 294
9.1 The Inverse Spectral Problem......Page 295
9.2 Complex Dimensions and the Riemann Hypothesis......Page 298
9.3 Fractal Sprays and the Generalized Riemann Hypothesis......Page 301
9.4 Notes......Page 303
10.1 The Geometry of a Generalized Cantor String......Page 305
10.2 The Spectrum of a Generalized Cantor String......Page 308
10.2.1 Integral Cantor Strings: a-adic Analysis of the Geometric and Spectral Oscillations......Page 310
10.3 The Truncated Cantor String......Page 313
10.3.1 The Spectrum of the Truncated Cantor String......Page 316
10.4 Notes......Page 317
11 Critical Zeros of Zeta Functions......Page 318
11.1 The Riemann Zeta Function: No Critical Zeros in Arithmetic Progression......Page 319
11.1.1 Finite Arithmetic Progressions of Zeros......Page 322
11.2 Extension to Other Zeta Functions......Page 328
11.3 Density of Nonzeros on Vertical Lines......Page 330
11.3.1 Almost Arithmetic Progressions of Zeros......Page 331
11.4 Extension to L-Series......Page 332
11.4.1 Finite Arithmetic Progressions of Zeros of L-Series......Page 333
The Direct Computation......Page 336
The Explicit Formula......Page 337
Zeros in Arithmetic Progression......Page 339
11.5 Conjectures about Zeros of Dirichlet Series......Page 341
11.6 Zeta Functions of Curves Over Finite Fields......Page 345
12 Fractality and Complex Dimensions......Page 354
12.1 A New Definition of Fractality......Page 355
12.1.1 Fractal Geometers’ Intuition of Fractality......Page 356
12.1.2 Our Definition of Fractality......Page 358
12.1.3 Possible Connections with the Notion of Lacunarity......Page 363
12.2 Fractality and Self-Similarity......Page 365
12.2.1 Complex Dimensions and Tube Formula for the von Koch Snowflake Curve......Page 367
12.3 Complex Dimensions and Shifts......Page 374
12.4 Complex Dimensions as Geometric Invariants......Page 375
12.4.1 Connection with Varieties over Finite Fields......Page 377
Fractal Geometries and Finite Geometries......Page 378
12.5.1 The Weyl–Berry Conjecture......Page 381
12.5.2 The Spectrum of a Self-Similar Drum......Page 382
12.5.3 Spectrum and Periodic Orbits......Page 386
12.6 Notes......Page 391
13 New Results and Perspectives......Page 393
13.1 A Higher-Dimensional Theory of Complex Dimensions......Page 394
13.1.1 The Distributional Tube Formula for Fractal Sprays with Monophase Generators......Page 397
13.1.2 The Canonical Self-Similar Tiling......Page 403
13.1.3 Convex Geometry and the Curvature Measures......Page 409
13.1.4 Examples of Tube Formulas for Fractal Sprays and Self-Similar Tilings......Page 413
13.1.5 Pointwise Tube Formula for Fractal Sprays......Page 418
13.2 p-Adic Fractal Strings......Page 430
13.2.1 p-Adic Numbers......Page 431
13.2.2 p-Adic Fractal Strings......Page 433
13.2.3 Inner Tube and its Volume......Page 437
13.2.4 Tube Formulas for p-Adic Fractal Strings......Page 442
13.2.5 Self-Similar p-Adic Fractal Strings......Page 446
13.2.6 The Geometric Zeta Function......Page 450
13.2.7 Periodicity of the Poles and the Zeros......Page 451
13.2.8 Exact Tube Formula for p-Adic Self-Similar Strings......Page 454
13.2.9 Tube Formula for the 2-Adic Fibonacci String......Page 457
13.2.10 The 3-Adic Cantor Set......Page 459
13.2.11 The Minkowski Dimension......Page 461
13.2.12 The Average Minkowski Content......Page 465
13.2.13 Concluding Comments......Page 466
13.3 Multifractal Analysis via Zeta Functions......Page 467
13.3.1 Regularity......Page 469
13.3.2 Multifractal Zeta Functions......Page 470
13.3.3 Results for Fractal Strings......Page 472
13.3.4 Variants of the Cantor String......Page 475
13.3.5 A Full Collection of Multifractal Zeta Functions......Page 478
13.3.6 A Classic Multifractal and Other Zeta Functions......Page 482
13.4 Random Fractal Strings and their Zeta Functions......Page 487
13.4.1 Random Self-Similar Strings......Page 490
13.4.2 Stable Random Strings......Page 491
13.4.3 A Refinement of the Notion of Fractality......Page 493
13.5 Fractal Membranes: Quantized Fractal Strings......Page 494
13.5.1 Noncommutative, Infinite Dimensional Tori......Page 495
Fractal Membranes and their Zeta Functions......Page 497
13.5.2 Construction of Fractal Membranes......Page 499
13.5.3 Flows on the Moduli Space of Fractal Membranes......Page 501
13.6 Notes......Page 503
A.1 The Dedekind Zeta Function......Page 504
A.2 Characters and Hecke L-series......Page 505
A.3 Completion of L-Series, Functional Equation......Page 506
A.4 Epstein Zeta Functions......Page 507
A.5 Two-Variable Zeta Functions......Page 508
A.5.1 The Zeta Function of Pellikaan......Page 509
A.5.2 The Zeta Function of Schoof and van der Geer......Page 511
A.6 Other Zeta Functions in Number Theory......Page 514
B.1 Weyl’s Asymptotic Formula......Page 515
B.2 Heat Asymptotic Expansion......Page 517
B.3 The Spectral Zeta Function and its Poles......Page 518
B.4 Extensions......Page 520
B.4.1 Monotonic Second Term......Page 521
B.5 Notes......Page 522
Appendix C An Application of Nevanlinna Theory......Page 523
C.1 The Nevanlinna Height......Page 524
C.2 Complex Zeros of Dirichlet Polynomials......Page 525
Acknowledgements......Page 529
Bibliography......Page 533
Author Index......Page 566
Subject Index......Page 569
Index of Symbols......Page 579
Conventions......Page 583