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

Lapidus, Michel L.; Van Frankenhuysen, Machiel

ISBN 10: 1461253160 / ISBN 13: 9781461253167

Edité par Birkhäuser, 2012

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N° de réf. du vendeur 18676475-n

Détails bibliographiques

• Titre: Fractal Geometry and Number Theory : Complex Dimensions of Fractal Strings and Zeros of Zeta Functions
• Éditeur: Birkhäuser
• Date de parution: 2012
• Langue: anglais
• ISBN à 10 chiffres: 1461253160
• ISBN à 13 chiffres: 9781461253167
• Reliure: Couverture souple
• État de l'article: New

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Synopsis

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.