Fractal Strings and Multifractal Zeta Functions

Abstract

For a Borel measure on the unit interval and a sequence of scales that tend to zero, we define a one-parameter family of zeta functions called multifractal zeta functions. These functions are a first attempt to associate a zeta function to certain multifractal measures. However, we primarily show that they associate a new zeta function, the topological zeta function, to a fractal string in order to take into account the topology of its fractal boundary. This expands upon the geometric information garnered by the traditional geometric zeta function of a fractal string in the theory of complex dimensions. In particular, one can distinguish between a fractal string whose boundary is the classical Cantor set, and one whose boundary has a single limit point but has the same sequence of lengths as the complement of the Cantor set. Later work will address related, but somewhat different, approaches to multifractals themselves, via zeta functions, partly motivated by the present paper.

arXiv comment32 pages, 9 figures. This revised version contains new sections and figures illustrating the main results of this paper and recent results from others. Sections 0, 2, and 6 have been significantly rewritten

Cited by in corpus (5)

• Fractal Zeta Functions and Complex Dimensions of Relative Fractal Drums
• Towards Quantized Number Theory: Spectral Operators and an Asymmetric Criterion for the Riemann Hypothesis
• Complex dimensions of fractals and meromorphic extensions of fractal zeta functions
• Minkowski measurability criteria for compact sets and relative fractal drums in Euclidean spaces
• Toward zeta functions and complex dimensions of multifractals