Distance and Tube Zeta Functions

Abstract

Distance and tube zeta functions of fractals in Euclidean spaces can be considered as a bridge between the geometry of fractal sets and the theory of holomorphic functions. This is first seen from their fundamental property: the upper box dimension of any bounded fractal is equal to the abscissa of convergence of its distance and tube zeta functions. Furthermore, under some natural conditions, the residue of the tube zeta function of a fractal, evaluated at its abscissa of convergence, is equal to its Minkowski content, a fractal analog of its volume. It is possible to obtain very general results dealing with the problem of meromorophic continuation of these two fractal zeta functions. We show, in particular, that the distance zeta function and the tube zeta function contain essentially the same information, both from the point of view of their meromorphic continuation (when it exists) to a given domain of the complex plane, of their poles (called visible complex dimensions) and their residues (or, more generally, their principal parts), which are related in a simple manner. Consequently, the higher-dimensional theory of complex dimensions can be developed by using either of these two fractal zeta functions, and much preferably, both of them since one of these zeta functions is often more natural or simpler to use in a given situation or example. A variety of examples are studied from this point of view throughout the book (including in this chapter, the (N − 1)-dimensional sphere, generalized Cantor sets and the a-string, and in later chapters, the N-dimensional analogs of the Sierpinski carpet and the Sierpinski gasket, as well as fractal nests, self-similar fractal sprays, two-parameter generalized Cantor sets, discrete and continuous spirals, geometric chirps, etc.). In the one-dimensional case (that is, in the case of fractal strings), we show that the geometric zeta function of a fractal string and the corresponding distance zeta function are equivalent (in a suitable sense), and, in fact, define the same complex dimensions (except possibly at s = 0); in particular, they have the same abscissa of convergence, equal to the upper Minkowski dimension of the fractal string (or, equivalently, of the associated fractal subset of the real line). As we shall see in later chapters, distance and tube zeta functions can also be viewed as a bridge to the transcendental theory of numbers. For these reasons, these new fractal zeta functions deserve to be seriously studied. In fact, as is suggested by the title of this research monograph, they are the central object of investigation in this chapter and, along with their poles (or ‘complex dimensions’), throughout the entire book.