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Books > Science & Mathematics > Mathematics > Applied mathematics > Fractals
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Fractal Geometry and Number Theory - Complex Dimensions of Fractal Strings and Zeros of Zeta Functions (Hardcover, 1999 ed.)
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Fractal Geometry and Number Theory - Complex Dimensions of Fractal Strings and Zeros of Zeta Functions (Hardcover, 1999 ed.)
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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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