Are galaxy distributions scale invariant? A perspective from dynamical systems theory

Unless there is an evidence for fractal scaling with a single exponent over distances 0.1 r 100 h−1 Mpc, then the widely accepted notion of scale invariance of the correlation integral for 0.1 r 10 h−1 Mpc must be questioned. The attempt to extract a scaling exponent ν from the correlation integral n(r) by plotting log(n(r)) vs. log(r) is unreliable unless the underlying point set is approximately monofractal. The extraction of a spectrum of generalized dimensions νq from a plot of the correlation integral generating function Gn(q) by a similar procedure is probably an indication that Gn(q) does not scale at all. We explain these assertions after defining the term multifractal, mutually inconsistent definitions having been confused together in the cosmology literature. Part of this confusion is traced to the confusion in interpreting a measure-theoretic formula written down by Hentschel and Procaccia in the dynamical systems theory literature, while other errors follow from confusing together entirely different definitions of multifractal from two different schools of thought. Most important are serious errors in data analysis that follow from taking for granted a largest term approximation that is inevitably advertised in the literature on both fractals and dynamical systems theory.