Singularities of the potential and energy integrals and scaling laws for the dimensions’ spectra

Abstract

The analiticity properties of the potential and generalized energy integrals, defined as the Mellin transforms, of the measure of a sphere and of the correlation integral, are investigated for the mixing repellers. The short-distance behaviour of the measure of a sphere and of the correlation integral is determined by the singularities of their Mellin transforms. For the linear Cantor systems all the singularities of the potential-energy integral at any fixed point are determined. Aggreement is found with the numerical results for the measure of a ball and thef(α) spectrum is analytically determined for an arbitrary number of scales. The generalized energy integral is found to be meromorphic for linear Cantor systems and three different cases can be distinguished: self-similar (equal scales), resonant (logarithms of the scales rationally dependent) and generic. Correspondingly we have periodic, asymptotically periodic and aperiodic oscillations affecting the leading power law in the short-distance behaviour of the correlation integral. The aperiodic behaviour occurs also in the nonlinear Cantor systems which can be approximated by sequences of linear Cantor systems. This picture is fully confirmed by numerical results.