Fractal zeta functions and complex dimensions of Ahlfors metric measure spaces

Abstract

While classical analysis dealt primarily with smooth spaces, much research has been done in the last half century on expanding the theory to the nonsmooth case. Metric Measure spaces are the natural setting for such analysis, and it is thus important to understand the geometry of subsets of these spaces. In this paper, we focus on the geometry of Ahlfors regular spaces, Metric Measure spaces with an additional regularity condition. Historically, fractals have been studied using different ideas of dimension which have all proven to be unsatisfactory to some degree. The theory of complex dimensions, developed by M. L. Lapidus and a number of collaborators, was developed in part to better understand fractality in the Euclidean case and seeks to overcome these problems. Of particular interest is the recent theory of complex dimensions in higher-dimensional Euclidean spaces, as studied by M. L. Lapidus, G. Radunović, and D. Žubrinić, who introduced and studied the properties of the distance and tube zeta functions, ${\zeta }_A$ and ${\tilde{\zeta }}_A$. We show that this theory of complex dimensions naturally generalizes to the case of Ahlfors regular spaces, as the distance and tube zeta functions can be modified to apply to these spaces and all of its main properties carry over. We also provide a selection of examples in Ahlfors spaces, as well as hints that the theory can be expanded to a more general setting.