Consistent scaling of multifractal measures: Multifractal spatial correlations

There are a number of apparently disparate problems in multifractal scaling whose solutions have remained unclear, ranging from rather pathological cases where the standard Legendre transformations do not produce effective measures for the Hölder exponent and Hausdorff-Besicovitch dimension to the problem of describing the scaling of point-point correlation functions of moments of multifractal measures. We prove that an equivalent statement of multifractal scaling is the invariance of the generating functions of the scaling transformation. We show that the invariance of the generating functions is what allows the moment integrals to scale with simple power laws. We show that this definition can be successfully extended to cover the scaling of point-point correlation functions of moments of multifractal measures. Previous attempts to solve this problem have lead to non-scale-invariant behavior, presented as an inconsistency by Cates and Deutsch [Phys. Rev. A 35, 4907 (1987)]. We propose that the invariance of generating functions under their own transformations is the central defining characteristic of scale invariance in multifractal scaling. © 1993 The American Physical Society.