Local complex dimensions of a fractal string

We investigate in this work a local version of the theory of fractal strings and associated geometric zeta functions. Such a generalization allows to describe the asymptotic behaviour of a “fractal” set in the neighborhood of any of its points. We give basic properties and several examples illustrating the possible range of situations concerning in particular the evolution of the local complex dimensions along the set and the relation between local and global zeta functions. 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 [7, 8, 9, 12, 13, 14, 5]. See also the book [10] and the more recent works [11, 15, 16]. 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 . For our purpose, no distinction needs to be made between the open set and the sequence of lengths of the intervals that define it. 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 . 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. To a fractal string is associated a complex function, the (geometric) zeta function of L: Definition 2. The geometric zeta function of a fractal string L is