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

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

von: Michel Lapidus, Machiel van Frankenhuijsen

Springer-Verlag, 2007

ISBN: 9780387352084 , 460 Seiten

Format: PDF

Kopierschutz: Wasserzeichen

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Preis: 69,99 EUR

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Mehr zum Inhalt

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


 

Contents

6

Preface

12

List of Figures

16

List of Tables

19

Overview

20

Introduction

23

Complex Dimensions of Ordinary Fractal Strings

31

1.1 The Geometry of a Fractal String

31

1.2 The Geometric Zeta Function of a Fractal String

38

1.3 The Frequencies of a Fractal String and the Spectral Zeta Function

45

1.4 Higher-Dimensional Analogue: Fractal Sprays

48

1.5 Notes

51

Complex Dimensions of Self- Similar Fractal Strings

54

2.1 Construction of a Self-Similar Fractal String

54

2.2 The Geometric Zeta Function of a Self- Similar String

59

2.3 Examples of Complex Dimensions of Self- Similar Strings

62

2.4 The Lattice and Nonlattice Case

72

2.5 The Structure of the Complex Dimensions

75

2.6 The Asymptotic Density of the Poles in the Nonlattice Case

82

2.7 Notes

83

Complex Dimensions of Nonlattice Self- Similar Strings: Quasiperiodic Patterns and Diophantine Approximation

84

3.1 Dirichlet Polynomial Equations

85

3.2 Examples of Dirichlet Polynomial Equations

87

3.3 The Structure of the Complex Roots

92

3.4 Approximating a Nonlattice Equation by Lattice Equations

99

3.5 Complex Roots of a Nonlattice Dirichlet Polynomial

111

3.6 Dimension-Free Regions

122

3.7 The Dimensions of Fractality of a Nonlattice String

129

3.8 A Note on the Computations

133

Generalized Fractal Strings Viewed as Measures

135

4.1 Generalized Fractal Strings

136

4.2 The Frequencies of a Generalized Fractal String

141

4.3 Generalized Fractal Sprays

146

4.4 The Measure of a Self-Similar String

146

4.5 Notes

151

Explicit Formulas for Generalized Fractal Strings

152

5.1 Introduction

152

5.2 Preliminaries: The Heaviside Function

157

5.3 Pointwise Explicit Formulas

161

5.4 Distributional Explicit Formulas

173

5.5 Example: The Prime Number Theorem

189

5.6 Notes

192

The Geometry and the Spectrum of Fractal Strings

194

6.1 The Local Terms in the Explicit Formulas

195

6.2 Explicit Formulas for Lengths and Frequencies

199

6.3 The Direct Spectral Problem for Fractal Strings

203

6.4 Self-Similar Strings

208

6.5 Examples of Non-Self-Similar Strings

217

6.6 Fractal Sprays

221

Periodic Orbits of Self-Similar Flows

227

7.1 Suspended Flows

228

7.2 Periodic Orbits, Euler Product

230

7.3 Self-Similar Flows

233

7.4 The Prime Orbit Theorem for Suspended Flows

239

7.5 The Error Term in the Nonlattice Case

244

7.6 Notes

248

Tubular Neighborhoods and Minkowski Measurability

250

8.1 Explicit Formulas for the Volume of Tubular Neighborhoods

251

8.2 Analogy with Riemannian Geometry

260

8.3 Minkowski Measurability and Complex Dimensions

261

8.4 Tube Formulas for Self-Similar Strings

266

8.5 Notes

281

The Riemann Hypothesis and Inverse Spectral Problems

284

9.1 The Inverse Spectral Problem

285

9.2 Complex Dimensions of Fractal Strings and the Riemann Hypothesis

288

9.3 Fractal Sprays and the Generalized Riemann Hypothesis

291

9.4 Notes

293

Generalized Cantor Strings and their Oscillations

295

10.1 The Geometry of a Generalized Cantor String

295

10.2 The Spectrum of a Generalized Cantor String

298

10.3 The Truncated Cantor String

303

10.4 Notes

307

The Critical Zeros of Zeta Functions

308

11.1 The Riemann Zeta Function: No Critical Zeros in Arithmetic Progression

309

11.2 Extension to Other Zeta Functions

318

11.3 Density of Nonzeros on Vertical Lines

320

11.4 Extension to L-Series

322

11.5 Zeta Functions of Curves Over Finite Fields

331

Concluding Comments, Open Problems, and Perspectives

340

12.1 Conjectures about Zeros of Dirichlet Series

342

12.2 A New Definition of Fractality

345

12.3 Fractality and Self-Similarity

355

12.4 Random and Quantized Fractal Strings

369

12.5 The Spectrum of a Fractal Drum

384

12.6 The Complex Dimensions as the Spectrum of Shifts

393

12.7 The Complex Dimensions as Geometric Invariants

393

12.8 Notes

399

Zeta Functions in Number Theory

402

A.1 The Dedekind Zeta Function

402

A.2 Characters and Hecke L-series

403

A.3 Completion of L-Series, Functional Equation

404

A.4 Epstein Zeta Functions

405

A.5 Two-Variable Zeta Functions

406

A.6 Other Zeta Functions in Number Theory

411

Zeta Functions of Laplacians and Spectral Asymptotics

413

B.1 Weyl’s Asymptotic Formula

413

B.2 Heat Asymptotic Expansion

415

B.3 The Spectral Zeta Function and its Poles

416

B.4 Extensions

418

B.5 Notes

420

An Application of Nevanlinna Theory

421

C.1 The Nevanlinna Height

422

C.2 Complex Zeros of Dirichlet Polynomials

423

Bibliography

427

Acknowledgements

452

Conventions

456

Index of Symbols

457

Author Index

461

Subject Index

463