Local complex dimensions of a fractal string

Abstract

We investigate in this work a local version of the theory of fractal strings and associated geometric zeta functions. Such a generalisation allows to describe the asymptotic behaviour of a ‘fractal’ set in the neighbourhood 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.