Mellin transforms of correlation integrals and generalized dimension of strange sets

Abstract

The Mellin transform of the correlation integral is introduced and proved to be equal to the energy integral whose divergence abscissa is a lower bound to the Hausdorff dimension. For some Julia sets exact results are obtained. For the linear Cantor sets on the real axis it is shown that the energy integral is meromorphic, and the real pole, determining the divergence abscissa, has a sequence of satellite poles equally spaced on a line parallel to the imaginary axis, which explain the oscillations observed in numerical calculations of the correlation integral. The order-d generalized energy integrals are introduced as Mellin transforms of the order-d correlation integrals and for the Cantor sets they are proved to have the same singularities as the ordinary energy integrals. Letting ${r}_{d}$ be the residue of the real pole corresponding to the divergence abscissa it is proved that ${\mathrm{lim}}_{\mathrm{d}\ensuremath{\rightarrow}\mathrm{\ensuremath{\infty}}}$(-${\mathrm{d}}^{\mathrm{\ensuremath{-}}1}$ln${r}_{d}$) is the second Renyi entropy. Some numerical results obtained for the energy integrals are discussed.