 | Lapidus M.L. Fractal geometry, complex dimensions, and zeta functions: geometry and spectra of fractal strings / M.L.Lapidus, M. van Frankenhuijsen. - 2nd ed. - New York: Springer, 2013. - xxv, 567 p.: ill. - (Springer monographs in mathematics). - Bibliogr.: p.515-547. - Auth. ind.: p.549-551. - Sub. ind.: p.553-562. - Ind. of symb.: p.563-566. - ISBN 978-1-4614-2175-7
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Overview ................................................... vii
Preface ..................................................... xi
List of Figures ............................................ xxi
List of Tables ............................................. xxv
Introduction ................................................. 1
1 Complex Dimensions of Ordinary Fractal Strings ............... 9
1.1 The Geometry of a Fractal String ........................ 9
1.1.1 The Multiplicity of the Lengths ................. 12
1.1.2 Example: The Cantor String ...................... 13
1.2 The Geometric Zeta Function of a Fractal String ........ 16
1.2.1 The Screen and the Window ....................... 19
1.2.2 The Cantor String (continued) ................... 22
1.3 The Frequencies and the Spectral Zeta Function ......... 24
1.4 Higher-Dimensional Analogue: Fractal Sprays ............ 27
1.5 Notes .................................................. 30
2 Complex Dimensions of Self-Similar Fractal Strings .......... 33
2.1 Construction of a Self-Similar Fractal String .......... 33
2.1.1 Relation with Self-Similar Sets ................. 35
2.2 The Geometric Zeta Function of a Self-Similar String ... 38
2.2.1 Self-Similar Strings with a Single Gap .......... 40
2.3 Examples of Complex Dimensions of Self-Similar
Strings ................................................ 41
2.3.1 The Cantor String ............................... 41
2.3.2 The Fibonacci String ............................ 43
2.3.3 The Modified Cantor and Fibonacci Strings ....... 46
2.3.4 A String with Multiple Poles .................... 47
2.3.5 Two Nonlattice Examples ......................... 49
2.4 The Lattice and Nonlattice Case ........................ 52
2.5 The Structure of the Complex Dimensions ................ 54
2.6 The Asymptotic Density of the Poles in the Nonlattice
Case ................................................... 61
2.7 Notes .................................................. 62
3 Complex Dimensions of Nonlattice Self-Similar Strings ....... 65
3.1 Dirichlet Polynomial Equations ......................... 66
3.1.1 The Generic Nonlattice Case ..................... 67
3.2 Examples of Dirichlet Polynomial Equations ............. 68
3.2.1 Generic and Nongeneric Nonlattice Equations ..... 68
3.2.2 The Complex Roots of the Golden Plus Equation ... 73
3.3 The Structure of the Complex Roots ..................... 75
3.4 Approximating a Nonlattice Equation by Lattice
Equations .............................................. 80
3.4.1 Diophantine Approximation ....................... 83
3.4.2 The Quasiperiodic Pattern of the Complex
Dimensions ...................................... 86
3.4.3 Application to Nonlattice Strings ............... 89
3.5 Complex Roots of a Nonlattice Dirichlet Polynomial ..... 92
3.5.1 Continued Fractions ............................. 93
3.5.2 Two Generators .................................. 94
3.5.3 More than Two Generators ....................... 100
3.6 Dimension-Free Regions ................................ 103
3.7 The Dimensions of Fractality of a Nonlattice String ... 110
3.7.1 The Density of the Real Parts .................. 111
3.8 A Note on the Computations ............................ 115
3.9 Notes ................................................. 117
4 Generalized Fractal Strings Viewed as Measures ............. 119
4.1 Generalized Fractal Strings ........................... 120
4.1.1 Examples of Generalized Fractal Strings ........ 122
4.2 The Frequencies of a Generalized Fractal String ....... 124
4.2.1 Completion of the Harmonic String: Euler
Product ........................................ 127
4.3 Generalized Fractal Sprays ............................ 129
4.4 The Measure of a Self-Similar String .................. 130
4.4.1 Measures with a Self-Similarity Property ....... 132
4.5 Notes ................................................. 135
5 Explicit Formulas for Generalized Fractal Strings .......... 137
5.1 Introduction .......................................... 137
5.1.1 Outline of the Proof ........................... 139
5.1.2 Examples ....................................... 140
5.2 Preliminaries: The Heaviside Function ................. 142
5.3 Pointwise Explicit Formulas ........................... 146
5.3.1 The Order of Growth of the Sum over the
Complex Dimensions ............................. 157
5.4 Distributional Explicit Formulas ...................... 158
5.4.1 Extension to More General Test Functions ....... 163
5.4.2 The Order of the Distributional Error Term ..... 168
5.5 Example: The Prime Number Theorem ..................... 174
5.5.1 The Riemann-von Mangoldt Formula ............... 176
5.6 Notes ................................................. 177
6 The Geometry and the Spectrum of Fractal Strings ........... 179
6.1 The Local Terms in the Explicit Formulas .............. 180
6.1.1 The Geometric Local Terms ...................... 180
6.1.2 The Spectral Local Terms ....................... 182
6.1.3 The Weyl Term .................................. 183
6.1.4 The Distribution хω logm x ..................... 183
6.2 Explicit Formulas for Lengths and Frequencies ......... 184
6.2.1 The Geometric Counting Function of a Fractal
String ......................................... 184
6.2.2 The Spectral Counting Function of a Fractal
String ......................................... 185
6.2.3 The Geometric and Spectral Partition
Functions ...................................... 187
6.3 The Direct Spectral Problem for Fractal Strings ....... 189
6.3.1 The Density of Geometric and Spectral States ... 189
6.3.2 The Spectral Operator and its Euler Product .... 190
6.4 Self-Similar Strings .................................. 194
6.4.1 Lattice Strings ................................ 195
6.4.2 Nonlattice Strings ............................. 198
6.4.3 The Spectrum of a Self-Similar String .......... 200
6.5 Examples of Non-Self-Similar Strings .................. 203
6.5.1 The α-String ................................... 203
6.5.2 The Spectrum of the Harmonic String ............ 206
6.6 Fractal Sprays ........................................ 207
6.6.1 The Sierpinski Drum ............................ 208
6.6.2 The Spectrum of a Self-Similar Spray ........... 212
7 Periodic Orbits of Self-Similar Flows ...................... 213
7.1 Suspended Flows ....................................... 214
7.1.1 The Zeta Function of a Dynamical System ........ 215
7.2 Periodic Orbits, Euler Product ........................ 217
7.3 Self-Similar Flows .................................... 219
7.3.1 Examples of Self-Similar Flows ................. 223
7.3.2 The Lattice and Nonlattice Case ................ 224
7.4 The Prime Orbit Theorem for Suspended Flows ........... 226
7.4.1 The Prime Orbit Theorem for Self-Similar
Flows .......................................... 228
7.4.2 Lattice Flows .................................. 228
7.4.3 Nonlattice Flows ............................... 229
7.5 The Error Term in the Nonlattice Case ................. 230
7.5.1 Two Generators ................................. 230
7.5.2 More Than Two Generators ....................... 232
7.6 Notes ................................................. 235
8 Fractal Tube Formulas ...................................... 237
8.1 Explicit Formulas for the Volume of Tubular
Neighborhoods ......................................... 238
8.1.1 The Pointwise Tube Formula ..................... 243
8.1.2 Example: The α-String .......................... 246
8.2 Analogy with Riemannian Geometry ...................... 247
8.3 Minkowski Measurability and Complex Dimensions ........ 248
8.4 Tube Formulas for Self-Similar Strings ................ 253
8.4.1 Generalized Cantor Strings ..................... 253
8.4.2 Lattice Self-Similar Strings ................... 256
8.4.3 The Average Minkowski Content .................. 261
8.4.4 Nonlattice Self-Similar Strings ................ 264
8.5 Notes ................................................. 268
9 Riemann Hypothesis and Inverse Spectral Problems ........... 271
9.1 The Inverse Spectral Problem .......................... 272
9.2 Complex Dimensions and the Riemann Hypothesis ......... 275
9.3 Fractal Sprays and the Generalized Riemann
Hypothesis ............................................ 278
9.4 Notes ................................................. 280
10 Generalized Cantor Strings and their Oscillations .......... 283
10.1 The Geometry of a Generalized Cantor String ........... 283
10.2 The Spectrum of a Generalized Cantor String ........... 286
10.2.1 Integral Cantor Strings ........................ 288
10.2.2 Nonintegral Cantor Strings ..................... 291
10.3 The Truncated Cantor String .......................... 291
10.3.1 The Spectrum of the Truncated Cantor String .... 294
10.4 Notes ................................................. 295
11 Critical Zeros of Zeta Functions ........................... 297
11.1 Riemann Zeta Function: Zeros in Arithmetic
Progression ........................................... 298
11.1.1 Finite Arithmetic Progressions of Zeros ........ 301
11.2 Extension to Other Zeta Functions ..................... 307
11.3 Density of Nonzeros on Vertical Lines ................. 309
11.3.1 Almost Arithmetic Progressions of Zeros ........ 310
11.4 Extension to .L-Series ............................... 311
11.4.1 Finite Arithmetic Progressions of Zeros of
L-Series ....................................... 312
11.5 Conjectures about Zeros of Dirichlet Series .......... 320
11.6 Zeta Functions of Curves Over Finite Fields ........... 324
12 Fractality and Complex Dimensions ......................... 333
12.1 A New Definition of Fractality ....................... 334
12.1.1 Fractal Geometers' Intuition of Fractality ..... 335
12.1.2 Our Definition of Fractality ................... 337
12.1.3 Possible Connections with the Notion of
Lacunarity ..................................... 342
12.2 Fractality and Self-Similarity ........................ 344
12.2.1 The von Koch Snowflake Curve ................... 346
12.3 Complex Dimensions and Shifts ......................... 353
12.4 Complex Dimensions as Geometric Invariants ............ 354
12.4.1 Connection with Varieties over Finite Fields ... 356
12.4.2 Fractal Cohomology of Self-Similar Strings ..... 357
12.5 The Spectrum of a Fractal Drum ........................ 360
12.5.1 The Weyl-Berry Conjecture ...................... 360
12.5.2 The Spectrum of a Self-Similar Drum ............ 361
12.5.3 Spectrum and Periodic Orbits ................... 365
12.6 Notes ................................................. 370
13 New Results and Perspectives ............................... 373
13.1 A Higher-Dimensional Theory of Complex Dimensions ..... 374
13.1.1 Fractal Sprays with Monophase Generators ....... 377
13.1.2 The Canonical Self-Similar Tiling .............. 383
13.1.3 Convex Geometry and the Curvature Measures ..... 389
13.1.4 Examples of Fractal Sprays and Self-Similar
Tilings ........................................ 393
13.1.5 The Pointwise Tube Formula ..................... 398
13.2 p-Adic Fractal Strings ................................ 410
13.2.1 p-Adic Numbers ................................. 411
13.2.2 p-Adic Fractal Strings ......................... 413
13.2.3 Inner Tube and its Volume ...................... 417
13.2.4 Tube Formulas for p-Adic Fractal Strings ....... 422
13.2.5 Self-Similar p-Adic Fractal Strings ............ 426
13.2.6 The Geometric Zeta Function .................... 430
13.2.7 Periodicity of the Poles and the Zeros ......... 431
13.2.8 Exact Tube Formula for p-Adic Self-Similar
Strings ........................................ 434
13.2.9 Tube Formula for the 2-Adic Fibonacci String ... 437
13.2.10 The 3-Adic Cantor Set ......................... 439
13.2.11 The Minkowski Dimension ....................... 441
13.2.12 The Average Minkowski Content ................. 445
13.2.13 Concluding Comments ........................... 446
13.3 Multifractal Analysis via Zeta Functions .............. 447
13.3.1 Regularity ..................................... 449
13.3.2 Multifractal Zeta Functions .................... 450
13.3.3 Results for Fractal Strings .................... 452
13.3.4 Variants of the Cantor String .................. 455
13.3.5 A Full Collection of Multifractal Zeta
Functions ...................................... 458
13.3.6 A Classic Multifractal and Other Zeta
Functions ...................................... 462
13.4 Random Fractal Strings and their Zeta Functions ....... 467
13.4.1 Random Self-Similar Strings .................... 470
13.4.2 Stable Random Strings .......................... 471
13.4.3 A Refinement of the Notion of Fractality ....... 473
13.5 Fractal Membranes: Quantized Fractal Strings .......... 474
13.5.1 Noncommutative, Infinite Dimensional Tori ...... 475
13.5.2 Construction of Fractal Membranes .............. 479
13.5.3 Flows on the Moduli Space of Fractal
Membranes ...................................... 481
13.6 Notes ................................................. 483
A Zeta Functions in Number Theory ............................ 485
A.l The Dedekind Zeta Function ............................ 485
A.2 Characters and Hecke L-series ......................... 486
A.3 Completion of L-Series, Functional Equation ........... 487
A.4 Epstein Zeta Functions ................................ 488
A.5 Two-Variable Zeta Functions ........................... 489
A.5.1 The Zeta Function of Pellikaan ................. 490
A.5.2 The Zeta Function of Schoof and van der Geer ... 492
A.6 Other Zeta Functions in Number Theory ................. 495
В Zeta Functions of Laplacians and Spectral Asymptotics ...... 497
B.l Weyl's Asymptotic Formula ............................. 497
B.2 Heat Asymptotic Expansion ............................. 499
B.3 The Spectral Zeta Function and its Poles .............. 500
B.4 Extensions ............................................ 502
B.4.1 Monotonic Second Term .......................... 503
B.5 Notes ................................................. 504
С An Application of Nevanlinna Theory ........................ 505
C.l The Nevanlinna Height ................................. 506
C.2 Complex Zeros of Dirichlet Polynomials ................ 507
Acknowledgements ........................................... 511
Bibliography ............................................... 515
Author Index ............................................... 549
Subject Index .............................................. 553
Index of Symbols ........................................... 563
Conventions ................................................ 567
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