2006/10/06 by Michel L. Lapidus, Lapidus, Michel L., Jacques Lévy Véhel +4
Mathematics · Physics and Astronomy · #11M41 #28A12 #28A80 (Primary) 28A75 (Secondary) #Advanced Mathematical Theories and Applications #FOS: Physical sciences #Mathematical Dynamics and Fractals #Mathematical Physics (math-ph) #Theoretical and Computational Physics #math-ph #math.MP #msc:11M41 #msc:28A12 #msc:28A75 #msc:28A80
paper · pdf · doi:10.48550/arxiv.math-ph/0610015
32 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
openalex publication_date 2006/10/06 · arxiv created 2009/02/09 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.