Multifractal and higher dimensional zeta functions

International audienceIn 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 denition involving gap lengths. This modified zeta function allows us to denfine 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 yield the usual multi-fractal spectrum of the measure. The presence of complex poles for this function 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^n, 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