Multifractal and higher-dimensional zeta functions

• Published in: Nonlinearity 2011-01, Vol.24 (1), p.259-276
• Main Authors: Véhel, Jacques Lévy, Mendivil, Franklin
• Format: Article
• Language: English
• Subjects: Algebra; Asymptotic properties; Counting; Exact sciences and technology; Fractal analysis; Fractals; Global analysis, analysis on manifolds; Mathematical analysis; Mathematical methods in physics; Mathematics; Measure and integration; Number theory; Oscillations; Other topics in mathematical methods in physics; Physics; Probability; Sciences and techniques of general use; Self-similarity; Spectra; Strings; Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds
• ISSN: 0951-7715; 1361-6544
• DOI: 10.1088/0951-7715/24/1/013

In this paper, we generalize the zeta fonction for a fractal string (as in Lapidus and Frankenhuijsen 2006 Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings (New York: Springer)) in several directions. We first modify the zeta function to be associated with a sequence of covers instead of the usual definition involving gap lengths. This modified zeta function allows us to define both a multifractal zeta function and a zeta function for higher-dimensional fractal sets. In the multifractal case, the critical exponents of the zeta function [zeta](q, s) yield the usual multifractal spectrum of the measure. The presence of complex poles for [zeta](q, s) indicates oscillations in the continuous partition function of the measure, and thus gives more refined information about the multi fractal spectrum of a measure. In the case of a self-similar set in [dbl-struck R]", the modified zeta function yields asymptotic information about both the 'box' counting function of the set and the n-dimensional volume of the epsilon -dilation of the set.