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Fractal Geometry and Number Theory

Fractal Geometry and Number Theory


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About the Book

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.


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Product Details
  • ISBN-13: 9780817640989
  • Publisher: Birkhauser
  • Publisher Imprint: Birkhauser
  • Edition: 1999
  • Language: English
  • Returnable: N
  • Spine Width: 18 mm
  • Weight: 607 gr
  • ISBN-10: 0817640983
  • Publisher Date: 10 Dec 1999
  • Binding: Hardback
  • Height: 239 mm
  • No of Pages: 268
  • Series Title: English
  • Sub Title: Complex Dimensions of Fractal Strings and Zeros of Zeta Functions
  • Width: 163 mm


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