Fractal Geometry and Number Theory: Complex Dimensions of Fractal Strings and Zeros of Zeta Functions

Naslovnica
Springer Science & Business Media, 2000 - Broj stranica: 268
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 one-dimensional 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 [Berrl-2, Lapl-4, LapPol-3, LapMal-2, HeLapl-2] 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 [Lapl-3, LapPol-3, LapMal-2, HeLapl-2]). 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.
 

Sadržaj

Introduction
1
Complex Dimensions of Ordinary Fractal Strings
7
Complex Dimensions of SelfSimilar Fractal Strings
23
Generalized Fractal Strings Viewed as Measures
55
Explicit Formulas for Generalized Fractal Strings
71
The Geometry and the Spectrum of Fractal Strings
111
Tubular Neighborhoods and Minkowski Measurability
143
The Riemann Hypothesis Inverse Spectral Problems
163
The Critical Zeros of Zeta Functions
181
Concluding Comments
197
A Zeta Functions in Number Theory
221
B Zeta Functions of Laplacians and Spectral Asymptotics
227
References
235
Conventions
253
List of Figures
265
Autorska prava

Generalized Cantor Strings and their Oscillations
173

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