MEROMORPHIC STRUCTURE OF THE MELLIN TRANSFORMS AND SHORT-DISTANCE BEHAVIOR OF CORRELATION INTEGRALS

The short-distance behavior of the measure of a sphere and of the correlation integral is determined, in the case of disconnected repellers, by scaling laws whose corrections are oscillating functions, periodic or aperiodic, depending on exact or approximate self-similarity of the measure. The Mellin transforms prove to be the correct analytic tool in order to investigate these corrections to scaling. It has been previously proved that they are meromorphic for linear Cantor sets and that the leading pole gives the correlation dimension in agreement with the results of the thermodynamic formalism. Here we show that the residues of these poles can also be computed to any desired accuracy with simple algorithms and that the knowledge of the singularity spectrum of the Mellin transforms provides the Fourier spectrum of the scaling correction for the self-similar measure and that it reproduces the damped oscillations in the generic case. The method applies to the nonlinear repellers such as the disconnected Julia sets by using an approximation theorem.