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Overview
A fractal drum is a bounded open subset of R. m with a fractal boundary. A difficult problem is to describe the relationship between the shape (geo metry) of the drum and its sound (its spectrum). In this book, we restrict ourselves to the onedimensional case of fractal strings, and their higher dimensional analogues, fractal sprays. We develop a theory of complex di mensions of a fractal string, and we study how these complex dimensions relate the geometry with the spectrum of the fractal string. We refer the reader to [Berrl2, Lapl4, LapPol3, LapMal2, HeLapl2] and the ref erences therein for further physical and mathematical motivations of this work. (Also see, in particular, Sections 7. 1, 10. 3 and 10. 4, along with Ap pendix B. ) In Chapter 1, we introduce the basic object of our research, fractal strings (see [Lapl3, LapPol3, LapMal2, HeLapl2]). A 'standard fractal string' is a bounded open subset of the real line. Such a set is a disjoint union of open intervals, the lengths of which form a sequence which we assume to be infinite. Important information about the geometry of . c is contained in its geometric zeta function (c(8) = L lj. j=l 2 Introduction We assume throughout that this function has a suitable meromorphic ex tension. The central notion of this book, the complex dimensions of a fractal string . c, is defined as the poles of the meromorphic extension of (c.
Editorial Reviews
From the Publisher
"This highly original selfcontained book will appeal to geometers, fractalists, mathematical physicists and number theorists, as well as to graduate students in these fields and others interested in gaining insight into these rich areas either for its own sake or with a view to applications. They will find it a stimulating guide, well written in a clear and pleasant style."–Mathematical Reviews (Review of First Edition)
"It is the reviewer’s opinion that the authors have succeeded in showing that the complex dimensions provide a very natural and unifying mathematical framework for investigating the oscillations in the geometry and the spectrum of a fractal string. The book is well written. The exposition is selfcontained, intelligent and well paced."
–Bulletin of the London Mathematical Society (Review of First Edition)
"The new approach and results on the important problems illuminated in this work will appeal to researchers and graduate students in number theory, fractal geometry, dynamical systems, spectral geometry, and mathematical physics."
–Simulation News Europe (Review of First Edition)
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Table of Contents
1 Complex Dimensions of Ordinary Fractal Strings. 1.1 The Geometry of a Fractal String. 1.1.1 The Multiplicity of the Lengths. 1.1.2 Example: The Cantor String. 1.2 The Geometric Zeta Function of a Fractal String. 1.2.1 The Screen and the Window. 1.2.2 The Cantor String (Continued). 1.3 The Frequencies of a Fractal String and the Spectral Zeta Function. 1.4 HigherDimensional Analogue: Fractal Sprays. 2 Complex Dimensions of SelfSimilar Fractal Strings. 2.1 The Geometric Zeta Function of a SelfSimilar String. 2.1.1 Dynamical Interpretation, Euler Product. 2.2 Examples of Complex Dimensions of SelfSimilar Strings. 2.2.1 The Cantor String. 2.2.2 The Fibonacci String. 2.2.3 A String with Multiple Poles. 2.2.4 Two Nonlattice Examples. 2.3 The Lattice and Nonlattice Case. 2.3.1 Generic Nonlattice Strings. 2.4 The Structure of the Complex Dimensions. 2.5 The Density of the Poles in the Nonlattice Case. 2.5.1 Nevanlinna Theory. 2.5.2 Complex Zeros of Dirichlet Polynomials. 2.6 Approximating a Fractal String and Its Complex Dimensions. 2.6.1 Approximating a Nonlattice String by Lattice Strings. 3 Generalized Fractal Strings Viewed as Measures. 3.1 Generalized Fractal Strings. 3.1.1 Examples of Generalized Fractal Strings. 3.2 The Frequencies of a Generalized Fractal String. 3.3 Generalized Fractal Sprays. 3.4 The Measure of a SelfSimilar String. 3.4.1 Measures with a SelfSimilarity Property. 4 Explicit Formulas for Generalized Fractal Strings. 4.1 Introduction. 4.1.1 Outline of the Proof. 4.1.2 Examples. 4.2 Preliminaries: The Heaviside Function. 4.3 The Pointwise Explicit Formulas. 4.3.1 The Order of the Sum over the Complex Dimensions. 4.4 The Distributional Explicit Formulas. 4.4.1 Alternate Proof of Theorem 4.12. 4.4.2 Extension to More General Test Functions. 4.4.3 The Order of the Distributional Error Term. 4.5 Example: The Prime Number Theorem. 4.5.1 The Riemannvon Mangoldt Formula. 5 The Geometry and the Spectrum of Fractal Strings. 5.1 The Local Terms in the Explicit Formulas. 5.1.1 The Geometric Local Terms. 5.1.2 The Spectral Local Terms. 5.1.3 The Weyl Term. 5.1.4 The Distribution x?logmx. 5.2 Explicit Formulas for Lengths and Frequencies. 5.2.1 The Geometric Counting Function of a Fractal String. 5.2.2 The Spectral Counting Function of a Fractal String. 5.2.3 The Geometric and Spectral Partition Functions. 5.3 The Direct Spectral Problem for Fractal Strings. 5.3.1 The Density of Geometric and Spectral States. 5.3.2 The Spectral Operator. 5.4 SelfSimilar Strings. 5.4.1 Lattice Strings. 5.4.2 Nonlattice Strings. 5.4.3 The Spectrum of a SelfSimilar String. 5.4.4 The Prime Number Theorem for Suspended Flows. 5.5 Examples of NonSelfSimilar Strings. 5.5.1 The aString. 5.5.2 The Spectrum of the Harmonic String. 5.6 Fractal Sprays. 5.6.1 The Sierpinski Drum. 5.6.2 The Spectrum of a SelfSimilar Spray. 6 Tubular Neighborhoods and Minkowski Measurability. 6.1 Explicit Formula for the Volume of a Tubular Neighborhood. 6.1.1 Analogy with Riemannian Geometry. 6.2 Minkowski Measurability and Complex Dimensions. 6.3 Examples. 6.3.1 SelfSimilar Strings. 6.3.2 The aString. 7 The Riemann Hypothesis, Inverse Spectral Problems and Oscillatory Phenomena. 7.1 The Inverse Spectral Problem. 7.2 Complex Dimensions of Fractal Strings and the Riemann Hypothesis. 7.3 Fractal Sprays and the Generalized Riemann Hypothesis. 8 Generalized Cantor Strings and their Oscillations. 8.1 The Geometry of a Generalized Cantor String. 8.2 The Spectrum of a Generalized Cantor String. 8.2.1 Integral Cantor Strings: aadic Analysis of the Geometric and Spectral Oscillations. 8.2.2 Nonintegral Cantor Strings: Analysis of the Jumps in the Spectral Counting Function. 9 The Critical Zeros of Zeta Functions. 9.1 The Riemann Zeta Function: No Critical Zeros in an Arithmetic Progression. 9.2 Extension to Other Zeta Functions. 9.2.1 Density of Nonzeros on Vertical Lines. 9.2.2 Almost Arithmetic Progressions of Zeros. 9.3 Extension to LSeries. 9.4 Zeta Functions of Curves Over Finite Fields. 10 Concluding Comments. 10.1 Conjectures about Zeros of Dirichlet Series. 10.2 A New Definition of Fractality. 10.2.1 Comparison with Other Definitions of Fractality... 10.2.2 Possible Connections with the Notion of Lacunarity. 10.3 Fractality and SelfSimilarity. 10.4 The Spectrum of a Fractal Drum. 10.4.1 The WeylBerry Conjecture. 10.4.2 The Spectrum of a SelfSimilar Drum. 10.4.3 Spectrum and Periodic Orbits. 10.5 The Complex Dimensions as Geometric Invariants. Appendices. A Zeta Functions in Number Theory. A.l The Dedekind Zeta Function. A.3 Completion of LSeries, Functional Equation. A.4 Epstein Zeta Functions. A.5 Other Zeta Functions in Number Theory. B Zeta Functions of Laplacians and Spectral Asymptotics. B.l Weyl’s Asymptotic Formula. B.2 Heat Asymptotic Expansion. B.3 The Spectral Zeta Function and Its Poles. B.4 Extensions. B.4.1 Monotonic Second Term. References. Conventions. Symbol Index. List of Figures. Acknowledgements.