Astronomy and Astrophysics – Astrophysics
Scientific paper
1997-03-07
Physica A309 (2002) 183-213
Astronomy and Astrophysics
Astrophysics
39 pages, Latex with 17 eps-files, using epsf.sty and a4wide.sty (included) <mccauley@ife.no>
Scientific paper
10.1016/S0378-4371(02)00615-5
Unless there is evidence for fractal scaling with a single exponent over distances .1 <= r <= 100 h^-1 Mpc then the widely accepted notion of scale invariance of the correlation integral for .1 <= r <= 10 h^-1 Mpc must be questioned. The attempt to extract a scaling exponent \nu 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 \nu_q from a plot of the correlation integral generating function G_n(q) by a similar procedure is probably an indication that G_n(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 a misleading speculation made earlier 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.
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