Multifractal and higher dimensional zeta functions

In this paper, we generalize the zeta function for a fractal string (as in [18]) 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 ζ(q, s) yield the usual multifractal spectrum of the measure. The presence of complex poles for ζ(q, s) indicate oscillations in the continuous partition function of the measure, and thus give more refined information about the multifractal spectrum of a measure. In the case of a self-similar set in R, the modified zeta function yields asymptotic information about both the “box” counting function of the set and the n-dimensional volume of the -dilation of the set. 1 Background and Motivations The theory of fractal strings has been developed over the past years by M. Lapidus and co-workers in a series of papers, including [6, 10, 12, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, 26, 27, 32]. See also the book [18]. A fractal string L is simply a bounded open subset Ω of R. Ω may be written as a disjoint union of intervals Ij = (aj , bj), i.e. Ω = ⋃∞ j=1 Ij . Often, no distinction is made between the open set and its sequence of lengths. By abuse of notation, we will thus speak of the fractal string as L = {`j}j=1, where `j are the lengths of the Ij . We will also denote by {ln}n=1 the distinct lengths of L with multiplicities mn. Thus, a possible definition of a fractal string is: Definition 1 A fractal string L is an at most countable, non-increasing sequence of lengths whose sum is finite. ∗Regularity Team, Inria, Parc Orsay Université, 4 rue J. Monod, 91893 Orsay Cedex France, [email protected] †Department of Mathematics and Statistics, Acadia University, 12 University Avenue, Wolfville, NS Canada B4P 2R6, [email protected] 1 in ria -0 05 38 95 6, v er si on 1 24 N ov 2 01 0 Author manuscript, published in "Nonlinearity 24, 1 (2011) 259-276" DOI : 10.1088/0951-7715/24/1/013