Complex Dimensions of Fractal Strings and Explicit Formulas for Geometric andSpectral Zeta-Functions

摘要

In this work, we develop a theory of 'complex dimensions' of fractal strings.These complex dimensions are defined as the poles of the corresponding (geometric or spectral) zeta-funtions. They describe the oscillations in the geometry or the (frequency) spectrum of a fractal string, by means of an 'explicit formula'. In this first note, we establish the pointwise and distributional explicit formulas, which should be considered as the basic tools of this theory. Then we apply these explicit formulas to construct the 'spectral operator', that expresses the spectrum in terms of the geometry of a fractal string. We close this paper with a detailed study of the complex dimensions of self-similar fractal strings. This provides a large class of examples to which our theory can be applied fruitfully.