The Hentschel-Procaccia Conjecture

Abstract

Extensive confusion exists and persists in the literature on dynamical systems theory, cosmology, and other fields over spectra of fractal dimensions. Entirely different generating functions have been treated as if they should yield identical spectra, or scaling exponents. An uncritical implicit assumption of universality of scaling exponents is made, even though there is no good theoretical reason to expect scaling exponents to be universal away from criticality. Expectations based on the unphysical and empirically inapplicable zero length scale limit (which would require infinite precision in data analysis) are often taken for granted. A source of confusing together different generating functions can be traced back to the Hentschel-Procaccia conjecture, which I prove to be wrong for the case of data analysis in the only applicable limit, that of finite precision. The two different definitions of 'multifractal' stated by Mandelbrot and by Halsey et are compared and contrasted.