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Über dieses Buch

Number theory, spectral geometry, and fractal geometry are interlinked in this in-depth study of the vibrations of fractal strings, that is, one-dimensional drums with fractal boundary.

Key Features of this Second Edition:

The Riemann hypothesis is given a natural geometric reformulation in the context of vibrating fractal strings

Complex dimensions of a fractal string, defined as the poles of an associated zeta function, are studied in detail, then used to understand the oscillations intrinsic to the corresponding fractal geometries and frequency spectra

Explicit formulas are extended to apply to the geometric, spectral, and dynamical zeta functions associated with a fractal

Examples of such explicit formulas include a Prime Orbit Theorem with error term for self-similar flows, and a geometric tube formula

The method of Diophantine approximation is used to study self-similar strings and flows

Analytical and geometric methods are used to obtain new results about the vertical distribution of zeros of number-theoretic and other zeta functions

Throughout, new results are examined and a new definition of fractality as the presence of nonreal complex dimensions with positive real parts is presented. The new final chapter discusses several new topics and results obtained since the publication of the first edition.

The significant studies and problems illuminated in this work may be used in a classroom setting at the graduate level. Fractal Geometry, Complex Dimensions and Zeta Functions, Second Edition will appeal to students and researchers in number theory, fractal geometry, dynamical systems, spectral geometry, and mathematical physics.

Inhaltsverzeichnis

1. Complex Dimensions of Ordinary Fractal Strings

Abstract
In this chapter, we recall some basic definitions pertaining to the notion of (ordinary) fractal string and introduce several new ones, the most important of which is the notion of complex dimension. We also give a brief overview of some of our results in this context by discussing the simple but illustrative example of the Cantor string. In the last section, we discuss fractal sprays, which are a higher-dimensional analogue of fractal strings.
Michel L. Lapidus, Machiel van Frankenhuijsen

2. Complex Dimensions of Self-Similar Fractal Strings

Abstract
Throughout this book, we use an important class of ordinary fractal strings, the self-similar fractal strings, to illustrate our theory. These strings are constructed in the usual way via contraction mappings. In this and the next chapter, we give a detailed analysis of the structure of the complex dimensions of such fractal strings
Michel L. Lapidus, Machiel van Frankenhuijsen

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

Abstract
The study of the complex dimensions of nonlattice self-similar strings is most naturally carried out in the more general setting of Dirichlet polynomials.
Michel L. Lapidus, Machiel van Frankenhuijsen

4. Generalized Fractal Strings Viewed as Measures

Abstract
In this chapter, we develop the notion of generalized fractal string, viewed as a measure on the half-line. This is more general than the notion of fractal string considered in Chapter 1 and in the earlier work on this subject (see the notes to Chapter 1).
Michel L. Lapidus, Machiel van Frankenhuijsen

5. Explicit Formulas for Generalized Fractal Strings

Abstract
In this chapter, we obtain pointwise and distributional explicit formulas for the counting functions of the lengths and of the frequencies of a fractal string. These explicit formulas express these counting functions as a sum over the visible complex dimensions W of the fractal string.
Michel L. Lapidus, Machiel van Frankenhuijsen

6. The Geometry and the Spectrum of Fractal Strings

Abstract
In this chapter we give various examples of explicit formulas for the counting function of the lengths and frequencies of (generalized) fractal strings and sprays.
Michel L. Lapidus, Machiel van Frankenhuijsen

7. Periodic Orbits of Self-Similar Flows

Abstract
In this chapter, we apply our explicit formulas to obtain an asymptotic expansion for the prime orbit counting function of suspended flows. The resulting formula involves a sum of oscillatory terms associated with the dynamical complex dimensions of the flow.
Michel L. Lapidus, Machiel van Frankenhuijsen

8. Fractal Tube Formulas

Abstract
In this chapter, we obtain (in Section 8.1) a distributional formula for the volume of the tubular neighborhoods of the boundary of a fractal string, called a tube formula. In Section 8.1.1, under more restrictive assumptions, we also derive a tube formula that holds pointwise. In Section 8.3, we then deduce from these formulas a new criterion for the Minkowski measurability of a fractal string, in terms of its complex dimensions.
Michel L. Lapidus, Machiel van Frankenhuijsen

9. Riemann Hypothesis and Inverse Spectral Problems

Abstract
In this chapter, we provide an alternative formulation of the Riemann hypothesis in terms of a natural inverse spectral problem for fractal strings. After stating this inverse problem in Section 9.1, we show in Section 9.2 that its solution is equivalent to the nonexistence of critical zeros of the Riemann zeta function on a given vertical line.
Michel L. Lapidus, Machiel van Frankenhuijsen

10. Generalized Cantor Strings and their Oscillations

Abstract
In this chapter, we analyze the oscillations in the geometry and the spectrum of the simplest type of generalized self-similar fractal strings. The complex dimensions of these generalized Cantor strings form a vertical arithmetic sequence D + inp (n Z).
Michel L. Lapidus, Machiel van Frankenhuijsen

11. Critical Zeros of Zeta Functions

Abstract
As we saw in Chapter 10, the complex dimensions of a generalized Cantor string form an arithmetic progression {D + inp} nZ, with 0 D 1 and p 0. In this chapter we use this fact to study arithmetic progressions of critical zeros of zeta functions.
Michel L. Lapidus, Machiel van Frankenhuijsen

12. Fractality and Complex Dimensions

Abstract
In this chapter we discuss some more philosophical aspects of our theory of complex dimensions. In Section 12.1, we propose a new definition of fractality, involving the notion of complex dimension.
Michel L. Lapidus, Machiel van Frankenhuijsen

13. New Results and Perspectives

Abstract
In this chapter we discuss new work motivated by the notion of complex dimension. Throughout, we also make numerous suggestions for the direction of future research related to, and naturally extending in various ways, the theory developed in this book. In several places, we also provide some additional background material that may be useful to the reader.
Michel L. Lapidus, Machiel van Frankenhuijsen

Backmatter

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